Euclidian Geometry Grade 10 to 12 (CAPS)

Size: px
Start display at page:

Download "Euclidian Geometry Grade 10 to 12 (CAPS)"

Transcription

1 Euclidian Geometry Grade 10 to 12 (CAPS) Compiled by Marlene Malan Prepared by Marlene Malan

2 CAPS DOCUMENT (Paper 2) Grade 10 Grade 11 Grade 12 (a) Revise basic results established in earlier grades. (b) Investigate line segments joining the midpoints of two sides of a triangle. (c) Properties of special quadrilaterals. (a) Investigate and prove theorems of the geometry of circles assuming results from earlier grades, together with one other result concerning tangents and radii of circles. (b) Solve circle geometry problems, providing reasons for statements when required. (c) Prove riders. (a) Revise earlier (Grade 9) work on the necessary and sufficient conditions for polygons to be similar. (b) Prove (accepting results established in earlier grades): that a line drawn parallel to one side of a triangle divides the other two sides proportionally (and the Mid-point Theorem as a special case of this theorem); that equiangular triangles are similar; that triangles with sides in proportion are similar; the Pythagorean Theorem by similar triangles; riders. REVISION FROM EARLIER GRADES SIMILARITY AAA or SSS CONGRUENCY SSS AAS SAS (included angle) RHS

3 PROPERTIES OF SPECIAL QUADRILATERALS PARALLELOGRAM Both pairs of opposite sides are parallel Both pairs of opposite side are equal Both pairs of opposite angles are equal Diagonals bisect each other RECTANGLE All properties of parallelogram PLUS: Both diagonals are equal in length All interior angles are equal to 90 RHOMBUS All properties of parallelogram PLUS: All sides are equal Diagonals bisect each other perpendicularly Diagonals bisect interior angles SQUARE All properties of a rhombus PLUS: All interior angles are 90 Diagonals are equal in length KITE Two pairs of adjacent sides are equal Diagonal between equal sides bisects other diagonal One pair of opposite angles are equal (unequal sides) Diagonal between equal sides bisects interior angles (is axis of symmetry) Diagonals intersect perpendicularly TRAPEZIUM One pair of opposite sides are parallel HOW TO PROVE THAT A QUADRILATERAL IS A PARALLELOGRAM Prove any ONE of the following: Prove that both pairs of opposite sides are parallel Prove that both pairs of opposite sides are equal Prove that both pairs of opposite angles are equal Prove that the diagonals bisect each other Prove that ONE pair of sides are equal and parallel

4 HOW TO PROVE THAT A PARALLLELOGRAM IS A RHOMBUS Prove ONE of the following: Prove that the diagonals bisect each other perpendicularly Prove that any two adjacent sides are equal in length TRIANGLES BETWEEN PARALLEL LINES The AREA of two triangles on the SAME (OR EQUAL) BASE between two parallel lines, are EQUAL. Area of = Area of MIDPOINT THEOREM The line segment joining the midpoints of two sides of a triangle, is parallel to the third side of the triangle and half the length of that side. ( Midpt Theorem ) If AD = DB and AE = EC, then DE ǁ BC and DE = BC CONVERSE OF MIDPOINT THEOREM If a line is drawn from the midpoint of one side of a triangle parallel to another side, that line will bisect the third side and will be half the length of the side it is parallel to. ( line through midpoint to 2nd side ) If AD = DB and DE ǁ BC, then AE = EC and DE = BC.

5 GRADE 11 GEOMETRY Note: THEOREMS OF WHICH PROOFS ARE EXAMINABLE ARE INDICATED WITH Theorem 1 Converse of Theorem 1 If AC = CB in circle O, then OC AB. If OC chord AB, then AC = BC. (line from centre to midpt of chord) (line from centre to chord) Theorem 2 The angle at the centre of a circle subtended by an arc/a chord is double the angle at the circumference subtended by the same arc/chord. AO B=2 AC B ( at centre = 2 at circumference ) Theorem 3 Converse of Theorem 3 The angle on the circumference subtended by If =90, then AB is the diameter the diameter, is a right angle. of the circle. The angle in a semi-circle is 90. ( s in semi circle OR (chord subtends 90 OR diameter subtends right angle) converse s in semi circle)

6 Theorem 4 Converse of Theorem 4 The angles on the circumference of a If a line segment subtends equal angles circle subtended by the same arc or at two other points, then these four points chord, are equal. lie on the circumference of a circle. ( s in the same seg) (line subtends equal s OR converse s in the same seg) Corollary of Theorem 4 Equal chords subtend equal Equal chords subtend equal angles Equal chords of equal circles angles at the circumference at the centre of the circle. subtend equal angles at the of the circle. circumference. (equal chords; equal s) (equal chords; equal s) (equal circles; equal chords; equal s) Theorem 5 Converse of Theorem 5 The opposite angles of a cyclic quadrilateral are supplementary. = (opp s of cyclic quad ) If the opposite angles of a quadrilateral are supplementary, then it is a cyclic quadrilateral. ( opp s quad sup OR converse opp s of cyclic quad )

7 Theorem 6 Converse of Theorem 6 The exterior angle of a cyclic quadrilateral If the exterior angle of a quadrilateral is equal is equal to the opposite interior angle. to the opposite interior angle, then it is a cyclic quadrilateral. (ext of cyclic quad ) (ext = int opp OR converse ext of cyclic quad) Theorem 7 Converse of Theorem 7 The tangent to a circle is perpendicular to the If a line is drawn perpendicularly to the radius radius at the point of tangency. through the point where the radius meets the circle, then this line is a tangent to the circle. ( tan radius OR ( line radius OR converse tan radius OR tan diameter ) converse tan diameter ) Theorem 8 If two tangents are drawn from the same point outside a circle, then they are equal in length. (tans from common pt OR Tans from same pt )

8 Theorem 9 (Tan chord theorem) Converse of Theorem 9 The angle between the tangent to a circle and a chord drawn from the point of tangency, is equal to the angle in the opposite circle segment. If a line is drawn through the endpoint of a chord to form an angle which is equal to the angle in the opposite segment, then this line is a tangent. ( tan chord theorem ) ( converse tan chord theorem OR between line and chord ) Acute angle Obtuse angle THREE WAYS TO PROVE THAT A QUADRILATERAL IS A CYCLIC QUADRILATERAL Prove that : one pair of opposite angles are supplementary the exterior angle is equal to the opposite interior angle two angles subtended by a line segment at two other vertices of the quadrilateral, are equal.

9 GRADE 12 GEOMETRY The Concept of Proportionality (Revision) A 6 cm B 4 cm C D 9 cm E 6 cm F AB : BC = 6 : 4 = 3 : 2 and DE : EF = 9 : 6 = 3 : 2 Although, AB : BC = DE : EF it does NOT mean that AB = DE, AC = DF or BC = EF. Theorem 1 Converse of Theorem 1 A line drawn parallel to one side of a triangle If a line divides two sides of a triangle proporthat intersects the other two sides, will divide tionally, then the line is parallel to the third the other two sides proportionally. side of the triangle. ( line one side of Δ ( line divides two sides of Δ in prop ) OR prop theorem; name lines ) If DE ǁ BC then = or AD : DB = AE : EC If = then DE ǁ BC. Theorem 2 (Midpoint Theorem) Converse of Theorem 2 (Special case of Theorem 1) The line segment joining the midpoints of two If a line is drawn from the midpoint of one sides of a triangle, is parallel to the third side side of a triangle parallel to another side, that of the triangle and half the length of that side. line will bisect the third side and will be half the length of the side it is parallel to. ( midpt theorem ) ( line through midpt to 2 nd side ) If AD = DB and AE = EC, then DE ǁ BC and DE = BC If AD = DB and DE ǁ BC, then AE = EC and DE = BC.

10 Theorem 3 Converse of Theorem 3 The corresponding sides of two equiangular triangles are proportional and consequently the triangles are similar. If the sides of two triangles are proportional, then the triangles are equiangular and consequently the triangles are similar. ( Δs OR equiangular Δs ) ( Sides of Δ in prop ) If then If then Theorem 4 The perpendicular drawn from the vertex of the right angle of a right-angled triangle, divides the triangle in two triangles which are similar to each other and similar to the original triangle. Corollaries of Theorem 4... Theorem 5 (The Theorem of Pythagoras) From the corollaries it can be proven that:

11 TIPS TO SOLVING GEOMETRY RIDERS READ-READ-READ the information next to the diagram thoroughly TRANSFER all given information to the DIAGRAM Look for KEYWORDS, e.g. TANGENT: What do the theorems say about tangents? CYCLIC QUADRILATERAL: What are the properties of a cyclic quad? NEVER ASSUME something! - Don t assume that a certain line is the DIAMETER of a circle unless it is clearly state or unless you can prove it - Don t assume that a point is the CENTRE of a circle unless it is clearly stated ( circle M means the circle with midpoint M ) Set yourself SECONDARY GOALS, e.g. - To prove that = (primary goal), first prove that = (secondary goal) and vice versa - To prove that line AC is a tangent (primary goal), first prove that the line is perpendicular to radius OB (secondary goal) AC is tangent - To prove that BC is the diameter of the circle (primary goal), first prove that = 90 (secondary goal) BC is the diameter of the circle For questions like: Prove that. Start with ONE PART. Move to the OTHER PART step-by-step stating reasons. Remember it has to be clear and logical to the reader! E.g. = ; = ; = ; =

12 GRADE 11 GEOMETRY SAMPLE QUESTIONS Question 1 AB and CD are two chords of the circle with centre O., AF = FB, OE = 4 cm, OF = 3 cm and AB = 8 cm. Calculate the length of CD. [8] Question 2 Question 3 O is the centre of the circle. STU is a tangent at T. BC = CT 105 and =40 Calculate, giving reasons, the size of: 2.1 (2) 2.2 (2) 2.3 (3) 2.4 (6) [13] 3.1 Write down with reasons four other angles which are equal to. (8) 3.2 Prove that ABC EDC. (4) 3.3 Prove that =. (2) [14] Question 4 O is the centre of the circle. BC = CD Express the following in terms of : 4.1 (2) 4.2 (3) 4.3 (4) [9]

13 Question 5 LOM is the diameter of circle LMT. The centre of the circle is O. TN is a tangent at T. Prove that: 5.1 MNPT is a cyclic quadrilateral. (3) 5.2 NP = NT (6) [9] Question 6 PA and PC are tangents to the circle at C and A. AD ǁ PC and PD intersects the circle at B. Prove that: 6.1 bisects (6) 6.2 (6) 6.3 (4) [16] Question 7 TA is a tangent to the circle. M is the centre of chord PT.. O is the centre of the circle. Prove that: 7.1 MTAR is a cyclic quadrilateral. (3) 7.2 PR = RT (4) 7.3 TR bisects PTA (4) 7.4 (4) [15]

14 GRADE 12 GEOMETRY SAMPLE QUESTIONS Example Given: : 2:3 and. Instruction: Determine the ratio of :. Solution: In : = But it was given that = = = = = In : = = :=15:8 Question 1 =22, 33, 15. Calculate the value of. [4] Question 2,= : 4:3 Determine the ratio :. [8] Question 3, : 1: Write down the values of : and :. (2) 3.2 Determine : (1) 3.3 Prove that. (6) [9]

15 Question 4 Given: Prove that. [4] Question 5 is inscribed in a circle.,= PR is the diameter of the circle. Prove that: 5.1 (2) 5.2 O is the centre of the circle (2) 5.3 BORT is a trapezium. (2) [6] Question 6 Given: :=5:4 : 5:2 S is the midpoint of AQ 6.1 Prove that 2 (8) 6.2 If, determine : (6) [14] Question 7 Rectangle DEFK is drawn inside right-angled ABC. Prove that: 7.1.=. (4) 7.2 :=: (4) 7.3 :=: (1) 7.4 = (3) [12]

16 Question 8 ABOC is a kite with = = Why is? (2) 8.2 Complete: = = =... (3) 8.3 Prove that = (3) 8.4 Prove that =. (2) 8.5 If = =, prove that = 2. (2) [12]

17 MIXED EXERCISES 1. In the diagram, TBD is a tangent to circles BAPC and BNKM at B. AKC is a chord of the larger circle and is also a tangent to the smaller circle at K. Chords MN and BK intersect at F. PA is produced to D. BMC, BNA and BFKP are straight lines. Prove that: a) MN ǁ CA b) is isosceles c) = d) DA is a tangent to the circle passing through points A, B and K. 2. In the diagram below, chord BA and tangent TC of circle ABC are produced to meet at R. BC is produced to P with RC=RP. AP is not a tangent. Prove that: a) ACPR is a cyclic quadrilateral. b) c) =. d).=. e) Hence prove that =. 3. In the diagram alongside, circles ACBN and AMBD Intersect at A and B. CB is a tangent to the larger circle at B. M is the centre of the smaller circle. CAD and BND are straight lines. Let = a) Determine the size of in terms of. b) Prove that: i) CB ǁ AN ii) AB is a tangent to circle ADN.

18 4. In the diagram below, O is the centre of circle ABCD. DC is extended to meet circle BODE at point E. OE cuts BC at F. Let =. a) Determine in terms of. b) Prove that: i) BE=EC ii) BE is NOT a tangent to circle ABCD. 5. In the diagram alongside, medians AM and CN of intersect at O. BO is produced to meet AC at P. MP and CN intersect in D. ORǁMP with R on AC. a) Calculate, giving reasons, the numerical value of. b) Use :=2:3, to calculate the numerical value of. 6. In the diagram, AD is the diameter of circle ABCD. AD is extended to meet tangent NCP in P. Straight line NB is extended to Q and intersect AC in M with Q on straight line ADP. AC NQ at M. a) Prove that NQ ǁ CD. b) Prove that ANCQ is a cyclic quadrilateral. c) i) Prove that. ii) Hence, complete: = d) Prove that =. e) If it is further given that PC=MC, prove that 1. =.

19 SOLUTIONS TO MIXED EXERCISE 1. a) = tan chord = = corr s = b) = alt s = tan chord is isosceles c ) = alt s = = s in same segment s in same segment = alt s= line to one side of But line to one side of d) = s in same segment = equal chords subt equal s = is a tangent to the circle through A, B and K 2. a) = s opp equal sides = ext of = tan chord = = both = ACPR is a cyclic quadrilateral (ext of quad) b) In and : c) = = = = s in same segment proven in 2 a ext of cyclic quad 3 rd of. from 2 b but = =. d) In and : = tan chord is common = 3 rd angle

20 e).=. = = from 2. b) RC=RP =. From 2.d) =.. =. =. 3. a) = = s opp equal sides =180 2 =2 sum s of b. i) at centre =2x circ (90 2 sum s of ext of cyclic quad corr s b. ii) = =2 tan chord AB is a tangent alt s betw line&chord 4. a) s in same segment s opp = sides sum s of =90 at centre =2x at circumference b. i) 90 ext of cyclic quad 180 (90 sum s of =90 In and : = =90 BF = FC FE is common BE = EC s on str line s s b. ii) =90 sum s of = BE is not a tangent

21 5. a) P is midpoint of AC medians concur AB PM midpt theorem In : = = line one side of = = b) In : = = = = = BP is a median line one side of 6. a) =90 in semi =90 AM NM corr s= b) = lines, corr s = tan chord = ANCQ is a cyclic quad s subt by same line segm c) i) In and : = is common tan chord = 3 rd c) ii) =. d) In and : = s in same segm = s in same segm = tan chord = s in same segm = 3 rd. e 1 =.. Pyth.

Chapter (Circle) * Circle - circle is locus of such points which are at equidistant from a fixed point in

Chapter (Circle) * Circle - circle is locus of such points which are at equidistant from a fixed point in Chapter - 10 (Circle) Key Concept * Circle - circle is locus of such points which are at equidistant from a fixed point in a plane. * Concentric circle - Circle having same centre called concentric circle.

More information

Part (1) Second : Trigonometry. Tan

Part (1) Second : Trigonometry. Tan Part (1) Second : Trigonometry (1) Complete the following table : The angle Ratio 42 12 \ Sin 0.3214 Cas 0.5321 Tan 2.0625 (2) Complete the following : 1) 46 36 \ 24 \\ =. In degrees. 2) 44.125 = in degrees,

More information

0811ge. Geometry Regents Exam

0811ge. Geometry Regents Exam 0811ge 1 The statement "x is a multiple of 3, and x is an even integer" is true when x is equal to 1) 9 ) 8 3) 3 4) 6 In the diagram below, ABC XYZ. 4 Pentagon PQRST has PQ parallel to TS. After a translation

More information

chapter 1 vector geometry solutions V Consider the parallelogram shown alongside. Which of the following statements are true?

chapter 1 vector geometry solutions V Consider the parallelogram shown alongside. Which of the following statements are true? chapter vector geometry solutions V. Exercise A. For the shape shown, find a single vector which is equal to a)!!! " AB + BC AC b)! AD!!! " + DB AB c)! AC + CD AD d)! BC + CD!!! " + DA BA e) CD!!! " "

More information

RMT 2013 Geometry Test Solutions February 2, = 51.

RMT 2013 Geometry Test Solutions February 2, = 51. RMT 0 Geometry Test Solutions February, 0. Answer: 5 Solution: Let m A = x and m B = y. Note that we have two pairs of isosceles triangles, so m A = m ACD and m B = m BCD. Since m ACD + m BCD = m ACB,

More information

0811ge. Geometry Regents Exam BC, AT = 5, TB = 7, and AV = 10.

0811ge. Geometry Regents Exam BC, AT = 5, TB = 7, and AV = 10. 0811ge 1 The statement "x is a multiple of 3, and x is an even integer" is true when x is equal to 1) 9 2) 8 3) 3 4) 6 2 In the diagram below, ABC XYZ. 4 Pentagon PQRST has PQ parallel to TS. After a translation

More information

I pledge that I have neither given nor received help with this assessment.

I pledge that I have neither given nor received help with this assessment. CORE MATHEMATICS PII Page 1 of 4 HILTON COLLEGE TRIAL EXAMINATION AUGUST 016 Time: 3 hours CORE MATHEMATICS PAPER 150 marks PLEASE READ THE FOLLOWING GENERAL INSTRUCTIONS CAREFULLY. 1. This question paper

More information

Name: Class: Date: 5. If the diagonals of a rhombus have lengths 6 and 8, then the perimeter of the rhombus is 28. a. True b.

Name: Class: Date: 5. If the diagonals of a rhombus have lengths 6 and 8, then the perimeter of the rhombus is 28. a. True b. Indicate whether the statement is true or false. 1. If the diagonals of a quadrilateral are perpendicular, the quadrilateral must be a square. 2. If M and N are midpoints of sides and of, then. 3. The

More information

Page 1 of 15. Website: Mobile:

Page 1 of 15. Website:    Mobile: Exercise 10.2 Question 1: From a point Q, the length of the tangent to a circle is 24 cm and the distance of Q from the centre is 25 cm. The radius of the circle is (A) 7 cm (B) 12 cm (C) 15 cm (D) 24.5

More information

Geometry Honors Review for Midterm Exam

Geometry Honors Review for Midterm Exam Geometry Honors Review for Midterm Exam Format of Midterm Exam: Scantron Sheet: Always/Sometimes/Never and Multiple Choice 40 Questions @ 1 point each = 40 pts. Free Response: Show all work and write answers

More information

Math 9 Chapter 8 Practice Test

Math 9 Chapter 8 Practice Test Name: Class: Date: ID: A Math 9 Chapter 8 Practice Test Short Answer 1. O is the centre of this circle and point Q is a point of tangency. Determine the value of t. If necessary, give your answer to the

More information

TRIANGLES CHAPTER 7. (A) Main Concepts and Results. (B) Multiple Choice Questions

TRIANGLES CHAPTER 7. (A) Main Concepts and Results. (B) Multiple Choice Questions CHAPTER 7 TRIANGLES (A) Main Concepts and Results Triangles and their parts, Congruence of triangles, Congruence and correspondence of vertices, Criteria for Congruence of triangles: (i) SAS (ii) ASA (iii)

More information

Geometry: Introduction, Circle Geometry (Grade 12)

Geometry: Introduction, Circle Geometry (Grade 12) OpenStax-CNX module: m39327 1 Geometry: Introduction, Circle Geometry (Grade 12) Free High School Science Texts Project This work is produced by OpenStax-CNX and licensed under the Creative Commons Attribution

More information

8. Quadrilaterals. If AC = 21 cm, BC = 29 cm and AB = 30 cm, find the perimeter of the quadrilateral ARPQ.

8. Quadrilaterals. If AC = 21 cm, BC = 29 cm and AB = 30 cm, find the perimeter of the quadrilateral ARPQ. 8. Quadrilaterals Q 1 Name a quadrilateral whose each pair of opposite sides is equal. Mark (1) Q 2 What is the sum of two consecutive angles in a parallelogram? Mark (1) Q 3 The angles of quadrilateral

More information

Geometry 3 SIMILARITY & CONGRUENCY Congruency: When two figures have same shape and size, then they are said to be congruent figure. The phenomena between these two figures is said to be congruency. CONDITIONS

More information

0114ge. Geometry Regents Exam 0114

0114ge. Geometry Regents Exam 0114 0114ge 1 The midpoint of AB is M(4, 2). If the coordinates of A are (6, 4), what are the coordinates of B? 1) (1, 3) 2) (2, 8) 3) (5, 1) 4) (14, 0) 2 Which diagram shows the construction of a 45 angle?

More information

Triangles. Example: In the given figure, S and T are points on PQ and PR respectively of PQR such that ST QR. Determine the length of PR.

Triangles. Example: In the given figure, S and T are points on PQ and PR respectively of PQR such that ST QR. Determine the length of PR. Triangles Two geometric figures having the same shape and size are said to be congruent figures. Two geometric figures having the same shape, but not necessarily the same size, are called similar figures.

More information

0809ge. Geometry Regents Exam Based on the diagram below, which statement is true?

0809ge. Geometry Regents Exam Based on the diagram below, which statement is true? 0809ge 1 Based on the diagram below, which statement is true? 3 In the diagram of ABC below, AB AC. The measure of B is 40. 1) a b ) a c 3) b c 4) d e What is the measure of A? 1) 40 ) 50 3) 70 4) 100

More information

Topic 2 [312 marks] The rectangle ABCD is inscribed in a circle. Sides [AD] and [AB] have lengths

Topic 2 [312 marks] The rectangle ABCD is inscribed in a circle. Sides [AD] and [AB] have lengths Topic 2 [312 marks] 1 The rectangle ABCD is inscribed in a circle Sides [AD] and [AB] have lengths [12 marks] 3 cm and (\9\) cm respectively E is a point on side [AB] such that AE is 3 cm Side [DE] is

More information

0113ge. Geometry Regents Exam In the diagram below, under which transformation is A B C the image of ABC?

0113ge. Geometry Regents Exam In the diagram below, under which transformation is A B C the image of ABC? 0113ge 1 If MNP VWX and PM is the shortest side of MNP, what is the shortest side of VWX? 1) XV ) WX 3) VW 4) NP 4 In the diagram below, under which transformation is A B C the image of ABC? In circle

More information

Plane geometry Circles: Problems with some Solutions

Plane geometry Circles: Problems with some Solutions The University of Western ustralia SHL F MTHMTIS & STTISTIS UW MY FR YUNG MTHMTIINS Plane geometry ircles: Problems with some Solutions 1. Prove that for any triangle, the perpendicular bisectors of the

More information

Label carefully each of the following:

Label carefully each of the following: Label carefully each of the following: Circle Geometry labelling activity radius arc diameter centre chord sector major segment tangent circumference minor segment Board of Studies 1 These are the terms

More information

Properties of the Circle

Properties of the Circle 9 Properties of the Circle TERMINOLOGY Arc: Part of a curve, most commonly a portion of the distance around the circumference of a circle Chord: A straight line joining two points on the circumference

More information

0610ge. Geometry Regents Exam The diagram below shows a right pentagonal prism.

0610ge. Geometry Regents Exam The diagram below shows a right pentagonal prism. 0610ge 1 In the diagram below of circle O, chord AB chord CD, and chord CD chord EF. 3 The diagram below shows a right pentagonal prism. Which statement must be true? 1) CE DF 2) AC DF 3) AC CE 4) EF CD

More information

MATHEMATICS: PAPER II MARKING GUIDELINES

MATHEMATICS: PAPER II MARKING GUIDELINES NATIONAL SENIOR CERTIFICATE EXAMINATION NOVEMBER 05 MATHEMATICS: PAPER II MARKING GUIDELINES Time: 3 hours 50 marks These marking guidelines are prepared for use by examiners and sub-examiners, all of

More information

6 CHAPTER. Triangles. A plane figure bounded by three line segments is called a triangle.

6 CHAPTER. Triangles. A plane figure bounded by three line segments is called a triangle. 6 CHAPTER We are Starting from a Point but want to Make it a Circle of Infinite Radius A plane figure bounded by three line segments is called a triangle We denote a triangle by the symbol In fig ABC has

More information

VAISHALI EDUCATION POINT (QUALITY EDUCATION PROVIDER)

VAISHALI EDUCATION POINT (QUALITY EDUCATION PROVIDER) BY:Prof. RAHUL MISHRA Class :- X QNo. VAISHALI EDUCATION POINT (QUALITY EDUCATION PROVIDER) CIRCLES Subject :- Maths General Instructions Questions M:9999907099,9818932244 1 In the adjoining figures, PQ

More information

SM2H Unit 6 Circle Notes

SM2H Unit 6 Circle Notes Name: Period: SM2H Unit 6 Circle Notes 6.1 Circle Vocabulary, Arc and Angle Measures Circle: All points in a plane that are the same distance from a given point, called the center of the circle. Chord:

More information

Proofs. by Bill Hanlon

Proofs. by Bill Hanlon Proofs by Bill Hanlon Future Reference To prove congruence, it is important that you remember not only your congruence theorems, but know your parallel line theorems, and theorems concerning triangles.

More information

1 st Preparatory. Part (1)

1 st Preparatory. Part (1) Part (1) (1) omplete: 1) The square is a rectangle in which. 2) in a parallelogram in which m ( ) = 60, then m ( ) =. 3) The sum of measures of the angles of the quadrilateral equals. 4) The ray drawn

More information

Similarity of Triangle

Similarity of Triangle Similarity of Triangle 95 17 Similarity of Triangle 17.1 INTRODUCTION Looking around you will see many objects which are of the same shape but of same or different sizes. For examples, leaves of a tree

More information

21. Prove that If one side of the cyclic quadrilateral is produced then the exterior angle is equal to the interior opposite angle.

21. Prove that If one side of the cyclic quadrilateral is produced then the exterior angle is equal to the interior opposite angle. 21. Prove that If one side of the cyclic quadrilateral is produced then the exterior angle is equal to the interior opposite angle. 22. Prove that If two sides of a cyclic quadrilateral are parallel, then

More information

Triangles. 3.In the following fig. AB = AC and BD = DC, then ADC = (A) 60 (B) 120 (C) 90 (D) none 4.In the Fig. given below, find Z.

Triangles. 3.In the following fig. AB = AC and BD = DC, then ADC = (A) 60 (B) 120 (C) 90 (D) none 4.In the Fig. given below, find Z. Triangles 1.Two sides of a triangle are 7 cm and 10 cm. Which of the following length can be the length of the third side? (A) 19 cm. (B) 17 cm. (C) 23 cm. of these. 2.Can 80, 75 and 20 form a triangle?

More information

Geometry Final Review. Chapter 1. Name: Per: Vocab. Example Problems

Geometry Final Review. Chapter 1. Name: Per: Vocab. Example Problems Geometry Final Review Name: Per: Vocab Word Acute angle Adjacent angles Angle bisector Collinear Line Linear pair Midpoint Obtuse angle Plane Pythagorean theorem Ray Right angle Supplementary angles Complementary

More information

Class IX Chapter 8 Quadrilaterals Maths

Class IX Chapter 8 Quadrilaterals Maths Class IX Chapter 8 Quadrilaterals Maths Exercise 8.1 Question 1: The angles of quadrilateral are in the ratio 3: 5: 9: 13. Find all the angles of the quadrilateral. Answer: Let the common ratio between

More information

Class IX Chapter 8 Quadrilaterals Maths

Class IX Chapter 8 Quadrilaterals Maths 1 Class IX Chapter 8 Quadrilaterals Maths Exercise 8.1 Question 1: The angles of quadrilateral are in the ratio 3: 5: 9: 13. Find all the angles of the quadrilateral. Let the common ratio between the angles

More information

Fill in the blanks Chapter 10 Circles Exercise 10.1 Question 1: (i) The centre of a circle lies in of the circle. (exterior/ interior) (ii) A point, whose distance from the centre of a circle is greater

More information

Nozha Directorate of Education Form : 2 nd Prep. Nozha Language Schools Ismailia Road Branch

Nozha Directorate of Education Form : 2 nd Prep. Nozha Language Schools Ismailia Road Branch Cairo Governorate Department : Maths Nozha Directorate of Education Form : 2 nd Prep. Nozha Language Schools Sheet Ismailia Road Branch Sheet ( 1) 1-Complete 1. in the parallelogram, each two opposite

More information

Downloaded from

Downloaded from Triangles 1.In ABC right angled at C, AD is median. Then AB 2 = AC 2 - AD 2 AD 2 - AC 2 3AC 2-4AD 2 (D) 4AD 2-3AC 2 2.Which of the following statement is true? Any two right triangles are similar

More information

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Student Name:

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Student Name: GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Wednesday, August 17, 2011 8:30 to 11:30 a.m., only Student Name: School Name: Print your name and the name of

More information

Circles. Exercise 9.1

Circles. Exercise 9.1 9 uestion. Exercise 9. How many tangents can a circle have? Solution For every point of a circle, we can draw a tangent. Therefore, infinite tangents can be drawn. uestion. Fill in the blanks. (i) tangent

More information

Indicate whether the statement is true or false.

Indicate whether the statement is true or false. PRACTICE EXAM IV Sections 6.1, 6.2, 8.1 8.4 Indicate whether the statement is true or false. 1. For a circle, the constant ratio of the circumference C to length of diameter d is represented by the number.

More information

Maharashtra Board Class X Mathematics - Geometry Board Paper 2014 Solution. Time: 2 hours Total Marks: 40

Maharashtra Board Class X Mathematics - Geometry Board Paper 2014 Solution. Time: 2 hours Total Marks: 40 Maharashtra Board Class X Mathematics - Geometry Board Paper 04 Solution Time: hours Total Marks: 40 Note: - () All questions are compulsory. () Use of calculator is not allowed.. i. Ratio of the areas

More information

A plane can be names using a capital cursive letter OR using three points, which are not collinear (not on a straight line)

A plane can be names using a capital cursive letter OR using three points, which are not collinear (not on a straight line) Geometry - Semester 1 Final Review Quadrilaterals (Including some corrections of typos in the original packet) 1. Consider the plane in the diagram. Which are proper names for the plane? Mark all that

More information

(D) (A) Q.3 To which of the following circles, the line y x + 3 = 0 is normal at the point ? 2 (A) 2

(D) (A) Q.3 To which of the following circles, the line y x + 3 = 0 is normal at the point ? 2 (A) 2 CIRCLE [STRAIGHT OBJECTIVE TYPE] Q. The line x y + = 0 is tangent to the circle at the point (, 5) and the centre of the circles lies on x y = 4. The radius of the circle is (A) 3 5 (B) 5 3 (C) 5 (D) 5

More information

Udaan School Of Mathematics Class X Chapter 10 Circles Maths

Udaan School Of Mathematics Class X Chapter 10 Circles Maths Exercise 10.1 1. Fill in the blanks (i) The common point of tangent and the circle is called point of contact. (ii) A circle may have two parallel tangents. (iii) A tangent to a circle intersects it in

More information

LLT Education Services

LLT Education Services 8. The length of a tangent from a point A at distance 5 cm from the centre of the circle is 4 cm. Find the radius of the circle. (a) 4 cm (b) 3 cm (c) 6 cm (d) 5 cm 9. From a point P, 10 cm away from the

More information

Grade 9 Circles. Answer the questions. For more such worksheets visit

Grade 9 Circles. Answer the questions. For more such worksheets visit ID : ae-9-circles [1] Grade 9 Circles For more such worksheets visit www.edugain.com Answer the questions (1) Two circles with centres O and O intersect at two points A and B. A line PQ is drawn parallel

More information

8-6. a: 110 b: 70 c: 48 d: a: no b: yes c: no d: yes e: no f: yes g: yes h: no

8-6. a: 110 b: 70 c: 48 d: a: no b: yes c: no d: yes e: no f: yes g: yes h: no Lesson 8.1.1 8-6. a: 110 b: 70 c: 48 d: 108 8-7. a: no b: yes c: no d: yes e: no f: yes g: yes h: no 8-8. b: The measure of an exterior angle of a triangle equals the sum of the measures of its remote

More information

Department of Mathematics

Department of Mathematics Department of Mathematics TIME: 3 Hours Setter: DS DATE: 03 August 2015 GRADE 12 PRELIM EXAMINATION MATHEMATICS: PAPER II Total marks: 150 Moderator: AM Name of student: PLEASE READ THE FOLLOWING INSTRUCTIONS

More information

2012 GCSE Maths Tutor All Rights Reserved

2012 GCSE Maths Tutor All Rights Reserved 2012 GCSE Maths Tutor All Rights Reserved www.gcsemathstutor.com This book is under copyright to GCSE Maths Tutor. However, it may be distributed freely provided it is not sold for profit. Contents angles

More information

BOARD QUESTION PAPER : MARCH 2016 GEOMETRY

BOARD QUESTION PAPER : MARCH 2016 GEOMETRY BOARD QUESTION PAPER : MARCH 016 GEOMETRY Time : Hours Total Marks : 40 Note: (i) Solve All questions. Draw diagram wherever necessary. (ii) Use of calculator is not allowed. (iii) Diagram is essential

More information

QUESTION BANK ON STRAIGHT LINE AND CIRCLE

QUESTION BANK ON STRAIGHT LINE AND CIRCLE QUESTION BANK ON STRAIGHT LINE AND CIRCLE Select the correct alternative : (Only one is correct) Q. If the lines x + y + = 0 ; 4x + y + 4 = 0 and x + αy + β = 0, where α + β =, are concurrent then α =,

More information

RMT 2014 Geometry Test Solutions February 15, 2014

RMT 2014 Geometry Test Solutions February 15, 2014 RMT 014 Geometry Test Solutions February 15, 014 1. The coordinates of three vertices of a parallelogram are A(1, 1), B(, 4), and C( 5, 1). Compute the area of the parallelogram. Answer: 18 Solution: Note

More information

Mathematics 2260H Geometry I: Euclidean geometry Trent University, Winter 2012 Quiz Solutions

Mathematics 2260H Geometry I: Euclidean geometry Trent University, Winter 2012 Quiz Solutions Mathematics 2260H Geometry I: Euclidean geometry Trent University, Winter 2012 Quiz Solutions Quiz #1. Tuesday, 17 January, 2012. [10 minutes] 1. Given a line segment AB, use (some of) Postulates I V,

More information

8-6. a: 110 b: 70 c: 48 d: a: no b: yes c: no d: yes e: no f: yes g: yes h: no

8-6. a: 110 b: 70 c: 48 d: a: no b: yes c: no d: yes e: no f: yes g: yes h: no Lesson 8.1.1 8-6. a: 110 b: 70 c: 48 d: 108 8-7. a: no b: yes c: no d: yes e: no f: yes g: yes h: no 8-8. b: The measure of an exterior angle of a triangle equals the sum of the measures of its remote

More information

UNIT-8 SIMILAR TRIANGLES Geometry is the right foundation of all painting, I have decided to teach its rudiments and principles to all youngsters eager for art. 1. ABC is a right-angled triangle, right-angled

More information

Chapter 8 Similar Triangles

Chapter 8 Similar Triangles Chapter 8 Similar Triangles Key Concepts:.A polygon in which all sides and angles are equal is called a regular polygon.. Properties of similar Triangles: a) Corresponding sides are in the same ratio b)

More information

Answer Key. 9.1 Parts of Circles. Chapter 9 Circles. CK-12 Geometry Concepts 1. Answers. 1. diameter. 2. secant. 3. chord. 4.

Answer Key. 9.1 Parts of Circles. Chapter 9 Circles. CK-12 Geometry Concepts 1. Answers. 1. diameter. 2. secant. 3. chord. 4. 9.1 Parts of Circles 1. diameter 2. secant 3. chord 4. point of tangency 5. common external tangent 6. common internal tangent 7. the center 8. radius 9. chord 10. The diameter is the longest chord in

More information

TENTH YEAR MATHEMATICS

TENTH YEAR MATHEMATICS The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION TENTH YEAR MATHEMATICS Thursday, January 26, 1989-1:1.5 to 4:1.5 p.m., only The last page of the booklet is the answer sheet. Fold

More information

MEMO MATHEMATICS: PAPER II

MEMO MATHEMATICS: PAPER II MEMO CLUSTER PAPER 2016 MATHEMATICS: PAPER II Time: 3 hours 150 marks PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY 1. This question paper consists of 28 pages and an Information Sheet of 2 pages(i-ii).

More information

C XZ if C XY > C YZ. Answers for the lesson Apply Properties of Chords. AC } DB therefore you can t show C BC > C CD. 4x x 1 8.

C XZ if C XY > C YZ. Answers for the lesson Apply Properties of Chords. AC } DB therefore you can t show C BC > C CD. 4x x 1 8. LESSON 10.3 Answers for the lesson Apply Properties of Chords Copyright Houghton Mifflin Harcourt Publishing Company. All rights reserved. Skill Practice 1. Sample answer: Point Y bisects C XZ if C XY

More information

4.! ABC ~ DEF,! AC = 6 ft, CB = 3 ft, AB = 7 ft, DF = 9 ft.! What is the measure of EF?

4.! ABC ~ DEF,! AC = 6 ft, CB = 3 ft, AB = 7 ft, DF = 9 ft.! What is the measure of EF? Name:!!!!!!!!!!!!! Geo(2) GEOMETRY (2) REVIEW FOR FINAL EXAM #2 1. If ABC is similar to ADE, then AB AD =? AE. Which replaces the? to make the statement true? A. AC!! B. AE!! C. DE!! D. BC 2. In ABC,

More information

Concurrency and Collinearity

Concurrency and Collinearity Concurrency and Collinearity Victoria Krakovna vkrakovna@gmail.com 1 Elementary Tools Here are some tips for concurrency and collinearity questions: 1. You can often restate a concurrency question as a

More information

MATHEMATICS: PAPER II Page 1 of 24 HILTON COLLEGE TRIAL EXAMINATION AUGUST 2014 MATHEMATICS: PAPER II GENERAL INSTRUCTIONS

MATHEMATICS: PAPER II Page 1 of 24 HILTON COLLEGE TRIAL EXAMINATION AUGUST 2014 MATHEMATICS: PAPER II GENERAL INSTRUCTIONS MATHEMATICS: PAPER II Page of 4 HILTON COLLEGE TRIAL EXAMINATION AUGUST 04 Time: 3 hours MATHEMATICS: PAPER II GENERAL INSTRUCTIONS 50 marks PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY.. This question

More information

C=2πr C=πd. Chapter 10 Circles Circles and Circumference. Circumference: the distance around the circle

C=2πr C=πd. Chapter 10 Circles Circles and Circumference. Circumference: the distance around the circle 10.1 Circles and Circumference Chapter 10 Circles Circle the locus or set of all points in a plane that are A equidistant from a given point, called the center When naming a circle you always name it by

More information

SHW 1-01 Total: 30 marks

SHW 1-01 Total: 30 marks SHW -0 Total: 30 marks 5. 5 PQR 80 (adj. s on st. line) PQR 55 x 55 40 x 85 6. In XYZ, a 90 40 80 a 50 In PXY, b 50 34 84 M+ 7. AB = AD and BC CD AC BD (prop. of isos. ) y 90 BD = ( + ) = AB BD DA x 60

More information

10. Circles. Q 5 O is the centre of a circle of radius 5 cm. OP AB and OQ CD, AB CD, AB = 6 cm and CD = 8 cm. Determine PQ. Marks (2) Marks (2)

10. Circles. Q 5 O is the centre of a circle of radius 5 cm. OP AB and OQ CD, AB CD, AB = 6 cm and CD = 8 cm. Determine PQ. Marks (2) Marks (2) 10. Circles Q 1 True or False: It is possible to draw two circles passing through three given non-collinear points. Mark (1) Q 2 State the following statement as true or false. Give reasons also.the perpendicular

More information

1 / 23

1 / 23 CBSE-XII-017 EXAMINATION CBSE-X-008 EXAMINATION MATHEMATICS Series: RLH/ Paper & Solution Code: 30//1 Time: 3 Hrs. Max. Marks: 80 General Instuctions : (i) All questions are compulsory. (ii) The question

More information

Ch 10 Review. Multiple Choice Identify the choice that best completes the statement or answers the question.

Ch 10 Review. Multiple Choice Identify the choice that best completes the statement or answers the question. Ch 10 Review Multiple Choice Identify the choice that best completes the statement or answers the question. 1. In the diagram shown, the measure of ADC is a. 55 b. 70 c. 90 d. 180 2. What is the measure

More information

Circles. II. Radius - a segment with one endpoint the center of a circle and the other endpoint on the circle.

Circles. II. Radius - a segment with one endpoint the center of a circle and the other endpoint on the circle. Circles Circles and Basic Terminology I. Circle - the set of all points in a plane that are a given distance from a given point (called the center) in the plane. Circles are named by their center. II.

More information

Core Mathematics 2 Coordinate Geometry

Core Mathematics 2 Coordinate Geometry Core Mathematics 2 Coordinate Geometry Edited by: K V Kumaran Email: kvkumaran@gmail.com Core Mathematics 2 Coordinate Geometry 1 Coordinate geometry in the (x, y) plane Coordinate geometry of the circle

More information

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Student Name:

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Student Name: GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Thursday, January 27, 2011 9:15 a.m. to 12:15 p.m., only Student Name: School Name: Print your name and the name

More information

MOCKTIME.COM ONLINE TEST SERIES CORRESPONDENCE COURSE

MOCKTIME.COM ONLINE TEST SERIES CORRESPONDENCE COURSE GEOMETRY TRIANGLES AND THEIR PROPERTIES A triangle is a figure enclosed by three sides. In the figure given below, ABC is a triangle with sides AB, BC, and CA measuring c, a, and b units, respectively.

More information

Class 9 Quadrilaterals

Class 9 Quadrilaterals ID : in-9-quadrilaterals [1] Class 9 Quadrilaterals For more such worksheets visit www.edugain.com Answer t he quest ions (1) The diameter of circumcircle of a rectangle is 13 cm and rectangle's width

More information

CO-ORDINATE GEOMETRY. 1. Find the points on the y axis whose distances from the points (6, 7) and (4,-3) are in the. ratio 1:2.

CO-ORDINATE GEOMETRY. 1. Find the points on the y axis whose distances from the points (6, 7) and (4,-3) are in the. ratio 1:2. UNIT- CO-ORDINATE GEOMETRY Mathematics is the tool specially suited for dealing with abstract concepts of any ind and there is no limit to its power in this field.. Find the points on the y axis whose

More information

NATIONAL SENIOR CERTIFICATE GRADE 11

NATIONAL SENIOR CERTIFICATE GRADE 11 NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS P NOVEMBER 05 MARKS: 50 TIME: 3 hours This question paper consists of 5 pages and a 4-page answer book. Mathematics/P DBE/November 05 CAPS Grade INSTRUCTIONS

More information

JUST IN TIME MATERIAL GRADE 11 KZN DEPARTMENT OF EDUCATION CURRICULUM GRADES DIRECTORATE TERM

JUST IN TIME MATERIAL GRADE 11 KZN DEPARTMENT OF EDUCATION CURRICULUM GRADES DIRECTORATE TERM JUST IN TIME MATERIAL GRADE 11 KZN DEPARTMENT OF EDUCATION CURRICULUM GRADES 10 1 DIRECTORATE TERM 1 017 This document has been compiled by the FET Mathematics Subject Advisors together with Lead Teachers.

More information

Definitions, Axioms, Postulates, Propositions, and Theorems from Euclidean and Non-Euclidean Geometries by Marvin Jay Greenberg ( )

Definitions, Axioms, Postulates, Propositions, and Theorems from Euclidean and Non-Euclidean Geometries by Marvin Jay Greenberg ( ) Definitions, Axioms, Postulates, Propositions, and Theorems from Euclidean and Non-Euclidean Geometries by Marvin Jay Greenberg (2009-03-26) Logic Rule 0 No unstated assumptions may be used in a proof.

More information

PRACTICE QUESTIONS CLASS IX: CHAPTER 4 LINEAR EQUATION IN TWO VARIABLES

PRACTICE QUESTIONS CLASS IX: CHAPTER 4 LINEAR EQUATION IN TWO VARIABLES PRACTICE QUESTIONS CLASS IX: CHAPTER 4 LINEAR EQUATION IN TWO VARIABLES 1. Find the value of k, if x =, y = 1 is a solution of the equation x + 3y = k.. Find the points where the graph of the equation

More information

Class IX - NCERT Maths Exercise (10.1)

Class IX - NCERT Maths Exercise (10.1) Class IX - NCERT Maths Exercise (10.1) Question 1: Fill in the blanks (i) The centre of a circle lies in of the circle. (exterior/interior) (ii) A point, whose distance from the centre of a circle is greater

More information

Exercise 10.1 Question 1: Fill in the blanks (i) The centre of a circle lies in of the circle. (exterior/ interior)

Exercise 10.1 Question 1: Fill in the blanks (i) The centre of a circle lies in of the circle. (exterior/ interior) Exercise 10.1 Question 1: Fill in the blanks (i) The centre of a circle lies in of the circle. (exterior/ interior) (ii) A point, whose distance from the centre of a circle is greater than its radius lies

More information

Honors Geometry Review Exercises for the May Exam

Honors Geometry Review Exercises for the May Exam Honors Geometry, Spring Exam Review page 1 Honors Geometry Review Exercises for the May Exam C 1. Given: CA CB < 1 < < 3 < 4 3 4 congruent Prove: CAM CBM Proof: 1 A M B 1. < 1 < 1. given. < 1 is supp to

More information

Mathematics 2260H Geometry I: Euclidean geometry Trent University, Fall 2016 Solutions to the Quizzes

Mathematics 2260H Geometry I: Euclidean geometry Trent University, Fall 2016 Solutions to the Quizzes Mathematics 2260H Geometry I: Euclidean geometry Trent University, Fall 2016 Solutions to the Quizzes Quiz #1. Wednesday, 13 September. [10 minutes] 1. Suppose you are given a line (segment) AB. Using

More information

Name: GEOMETRY: EXAM (A) A B C D E F G H D E. 1. How many non collinear points determine a plane?

Name: GEOMETRY: EXAM (A) A B C D E F G H D E. 1. How many non collinear points determine a plane? GMTRY: XM () Name: 1. How many non collinear points determine a plane? ) none ) one ) two ) three 2. How many edges does a heagonal prism have? ) 6 ) 12 ) 18 ) 2. Name the intersection of planes Q and

More information

Year 9 Term 3 Homework

Year 9 Term 3 Homework Yimin Math Centre Year 9 Term 3 Homework Student Name: Grade: Date: Score: Table of contents 5 Year 9 Term 3 Week 5 Homework 1 5.1 Geometry (Review)................................... 1 5.1.1 Angle sum

More information

X- MATHS IMPORTANT FORMULAS SELF EVALUVATION 1. SETS AND FUNCTIONS. 1. Commutative property i ii. 2. Associative property i ii

X- MATHS IMPORTANT FORMULAS SELF EVALUVATION 1. SETS AND FUNCTIONS. 1. Commutative property i ii. 2. Associative property i ii X- MATHS IMPORTANT FORMULAS SELF EVALUVATION 1. SETS AND FUNCTIONS 1. Commutative property i ii 2. Associative property i ii 3. Distributive property i ii 4. De Morgan s laws i ii i ii 5. Cardinality of

More information

Higher Geometry Problems

Higher Geometry Problems Higher Geometry Problems (1 Look up Eucidean Geometry on Wikipedia, and write down the English translation given of each of the first four postulates of Euclid. Rewrite each postulate as a clear statement

More information

Honors Geometry Mid-Term Exam Review

Honors Geometry Mid-Term Exam Review Class: Date: Honors Geometry Mid-Term Exam Review Multiple Choice Identify the letter of the choice that best completes the statement or answers the question. 1. Classify the triangle by its sides. The

More information

0110ge. Geometry Regents Exam Which expression best describes the transformation shown in the diagram below?

0110ge. Geometry Regents Exam Which expression best describes the transformation shown in the diagram below? 0110ge 1 In the diagram below of trapezoid RSUT, RS TU, X is the midpoint of RT, and V is the midpoint of SU. 3 Which expression best describes the transformation shown in the diagram below? If RS = 30

More information

SSC CGL Tier 1 and Tier 2 Program

SSC CGL Tier 1 and Tier 2 Program Gurudwara Road Model Town, Hisar 9729327755 www.ssccglpinnacle.com SSC CGL Tier 1 and Tier 2 Program -------------------------------------------------------------------------------------------------------------------

More information

Grade 9 Circles. Answer t he quest ions. For more such worksheets visit

Grade 9 Circles. Answer t he quest ions. For more such worksheets visit ID : th-9-circles [1] Grade 9 Circles For more such worksheets visit www.edugain.com Answer t he quest ions (1) ABCD is a cyclic quadrilateral such that AB is a diameter of the circle circumscribing it

More information

Geometry - Semester 1 Final Review Quadrilaterals

Geometry - Semester 1 Final Review Quadrilaterals Geometry - Semester 1 Final Review Quadrilaterals 1. Consider the plane in the diagram. Which are proper names for the plane? Mark all that apply. a. Plane L b. Plane ABC c. Plane DBC d. Plane E e. Plane

More information

Higher Geometry Problems

Higher Geometry Problems Higher Geometry Problems (1) Look up Eucidean Geometry on Wikipedia, and write down the English translation given of each of the first four postulates of Euclid. Rewrite each postulate as a clear statement

More information

1 / 24

1 / 24 CBSE-XII-017 EXAMINATION CBSE-X-01 EXAMINATION MATHEMATICS Paper & Solution Time: 3 Hrs. Max. Marks: 90 General Instuctions : 1. All questions are compulsory.. The question paper consists of 34 questions

More information

Collinearity/Concurrence

Collinearity/Concurrence Collinearity/Concurrence Ray Li (rayyli@stanford.edu) June 29, 2017 1 Introduction/Facts you should know 1. (Cevian Triangle) Let ABC be a triangle and P be a point. Let lines AP, BP, CP meet lines BC,

More information

Individual Events 1 I2 x 0 I3 a. Group Events. G8 V 1 G9 A 9 G10 a 4 4 B

Individual Events 1 I2 x 0 I3 a. Group Events. G8 V 1 G9 A 9 G10 a 4 4 B Answers: (99-95 HKMO Final Events) Created by: Mr. Francis Hung Last updated: July 08 I a Individual Events I x 0 I3 a I r 3 I5 a b 3 y 3 b 8 s b c 3 z c t 5 c d w d 0 u d 6 3 6 G6 a 5 G7 a Group Events

More information

Maharashtra State Board Class X Mathematics - Geometry Board Paper 2016 Solution

Maharashtra State Board Class X Mathematics - Geometry Board Paper 2016 Solution Maharashtra State Board Class X Mathematics - Geometry Board Paper 016 Solution 1. i. ΔDEF ΔMNK (given) A( DEF) DE A( MNK) MN A( DEF) 5 5 A( MNK) 6 6...(Areas of similar triangles) ii. ΔABC is 0-60 -90

More information

SOLUTIONS SECTION A [1] = 27(27 15)(27 25)(27 14) = 27(12)(2)(13) = cm. = s(s a)(s b)(s c)

SOLUTIONS SECTION A [1] = 27(27 15)(27 25)(27 14) = 27(12)(2)(13) = cm. = s(s a)(s b)(s c) 1. (A) 1 1 1 11 1 + 6 6 5 30 5 5 5 5 6 = 6 6 SOLUTIONS SECTION A. (B) Let the angles be x and 3x respectively x+3x = 180 o (sum of angles on same side of transversal is 180 o ) x=36 0 So, larger angle=3x

More information

Chapter 3. - parts of a circle.

Chapter 3. - parts of a circle. Chapter 3 - parts of a circle. 3.1 properties of circles. - area of a sector of a circle. the area of the smaller sector can be found by the following formula: A = qº 360º pr2, given q in degrees, or!

More information