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


 Ernest Brooks
 4 years ago
 Views:
Transcription
1 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 (i) three vertices namely, A,B and C (ii) three sides namely, AB, BC, CA (iii) three angles namely, A, B and C Types of on the basis of sides (i) Equilateral triangle A triangle whose all the three sides are equal is called equilateral In the figure ABC is an equilateral triangle in which AB = BC = CA (ii) Isosceles triangle A triangle having two sides equal is called an isosceles triangle In the figure, ABC is an isosceles triangle in which AB = AC (iii) Scalene triangle A triangle whose sides are of different lengths In the figure ABC is a triangle in which AB BC CA Types of on the Basis of Angles (i) Obtuseangled triangle A triangle in which one angle is an obtuse angle, is called an obtuse angled triangle In figure, ABC is a triangle in which B 9 (ii) Acute angle triangle a triangle in which all angles are less than 9 in measures is called acute angled triangle /
2 (iii) A right angled triangle : A triangle in which one angle is of exact 9 is called right angle triangle Some Other Important Terms of Median A median of a triangle is the line segment joining the midpoint of side with the opposite vertex (b) Centroid The point of intersection of all the three medians of a triangle is called its centroid Characteristics of Centroid (i) Centroid is the point at which the medians of triangle meet (ii) The medians of a triangle are concurrent (iii) The centroid divides the medians in the ratio 2 : 1 (iv) The median of an equilateral triangle are equal (v) The medians of an equilateral triangle coincide with the altitudes Altitudes The altitude of a triangle is the perpendicular drawn from a vertex to the opposite side (d) Orthocentre The point of intersection of all the three altitudes of a triangle is called its orthocenter Characteristics of Orthocentre (i) Orthocentre is the point at which the altitudes of a triangle meet (ii) The altitudes of a triangle are concurrent (iii) Orthocentre of an acute triangle lies in the interior of the triangle (iv) Orthocentre of an obtuse triangle lies in the exterior of the triangle Orthocentre of a right triangle lies on the vertex of the right angle (e) (f) Angle bisectot: the angle bisector of an angle of a triangle is a line that divided the angle in two equal part Incentre of a triangle The point of intersection of the bisectors the internal angles of a triangle in called its incentre Characteristics of Incentre (i) The point at which the three angle bisectors of a triangle intersect is called the incentre /
3 (ii) The triangle may be acute, obtuse, or right, the angle bisectors of a triangle must meet at a point lying inside the triangle (iii) The incentre of a triangle lies in the interior of the triangle (iv) The bisectors of the angles of a triangle are concurrent (v) From the incentre we can draw a on opposite sides (vi) We can call this perpendicular as inradius (g) Perpendicular bisector: the perpendicular bisector of a triangle is perpendicular drawn from the opposite vertex and divide the opposite side in two equal parts (h) Circumcentre of a triangle The point of intersection of the perpendicular bisectors of the sides of a triangle is called its circumcentre Characteristics of Circumcentre (i) The point at which the perpendicular bisectors of the sides of a triangle meet is called the cicumkcentre of the triangle (ii) The right bisectors of the sides of a triangle are concurrent (iii) With circumcentre as centre, we can drawn a circle passing through the vertices of a triangle (iv) The distance from centre to the vertices is called the circumradius The circle thus drawn with circumentre as centre and circumradius as radius is called circumcircle (i) Perimeter The sum of lengths of the sides of a figure is its perimeter Perimeter of ABC = AB + BC + CA Asseingment1 1 Prove that the sum of the angles of a triangle is 18 2 In ABC, B 75, C 32 find A 3 In the figure, show that A B C D E F /
4 4 In the figure, prove that a b c 36 5 The angles of a triangle are in the ratio 3 : 5 : 1 Find the measures of each angle of the triangle 6 The sum and difference of two angles of a triangle are 128 and 22 respectively Find all the angles of the triangle 7 If the bisector of the angle B and C of a ABC meet at a point O, then prove that BOC = A In ABC, B > C, if AM is the bisector of BAC and AN BC, prove that MAN = ( 2 B C ) 9 Fill in the blanks The sum of three angles of a triangle is (b) If two angles of a triangle are 51 and 38, the third angle is equal to If the angles of a triangle in the ratio 2 : 2 : 5, then the angles are (d) The angles of a triangle are 3x 5, 2x + 55 and 5x 5 degrees then x is equal to (e) A triangle cannot have more than right angles (f) A triangle cannot have more than obtuse angle 1 Which of the following statements are true or false? An exterior angle of a triangle is less than either of its interior opposite angles (b) Sum of the three angles of a triangle is 18 A triangle can have two right angles (d) A triangle can have two acute angles (e) A triangle can have two obtuse angles (f) An exterior angle of a triangle is equal to the sum of the two interior opposite angles Exterior angle of a Triangle: Definition If the side BC of a triangle ABC is produced to ray BD, then ACD is called an exterior angle of triangle ABC at C, and is denoted by exterior ACD A and B are called remote interior angles or interior opposite angles Note : At each vertex there are two exterior angles Theorem Given A To prove Proof If a side of a triangle is produced, then the exterior angle so formed is equal to the sum of the two interior opposite angles ABC whose side BC has been produced to D forming exterior angle ACD A B ACD In ABC B ACB 18 A (1) ACB ACD 18 (2) (straight angle) From (1) and (2), we get ACB ACD A B ACB ACD A B /
5 A B ACD IX ACADEMIC QUESTIONS Subjective Assengment2 1 In the figure, find BED 2 In figure, prove that x A B C 3 In the figure, find x and y, if AB DF and AD FG 4 In the figure, prove that DE BF 5 In the figure, AB DG, AC DE, EDH = 25 and BAC = 2, Find x and y 6 In the figure, CD AB, ABE = 13 and BAC = 7 Find x and y /
6 7 If ABD = 125 and ACE = 13, then BAC = Congruent Figures The geometric figures are said to be congruent if they are exactly of the same shape and size Congruent segments Two segments are congruent if they are of the same length and conversely Hence in Fig AB CD A B C D (b) Congruent angles Two angles are congruent if they have equal measures and conversely Hence in fig BAC FDE Congruent circles Two circles are congruent if they have equal radii and conversely Hence in fig If r 1 = r 2 Then c 1 circle c 2 circle Rules for Congruent triangles Rules 1 (SAS) When two sides and the included angle are given In ABC and DEF If AB = DE, A D, AC = DF Then ABC DEF /
7 Rule 2 (SSS) When three sides are given In ABC and DEF If AB = DE, BC = EF, CA = FD Then ABC DEF Rule 3 (ASA) When two angles and the included side is given In ABC and DEF If B E, BC = EF, C F Then ABC DEF Rule 4 (RHS) When Right Angle Hypotenuse Side are given In ABC and DEF If B E = 9, BC = EF, CA = FD Then ABC DEF Congruence Relations of (i) Reflexive ABC ABC (congruence relation is reflexive) Every triangle is congruent to itself (ii) Commutative If ABC DEF, then DEF ABC (congruence relation is commutative) (iii) Transitive If ABC DEF, and DEF PAR then ABC PAR (congruence relation is transitive) Note : Since the congruence relation is reflexive, commutative and transitive, it is an equivalence relation /
8 IX ACADEMIC QUESTIONS Subjective Asseingment3 1 Prove that ABC is isosceles if altitude AD bisects BC 2 Prove that ABC is isosceles if median AD is perpendicular to BC 3 In the fig it is given that AB = CF, EF = BD and AFE = DBC Prove that AFE CBD 4 ABCD is a quadrilateral in which AD = BC and DAB CBA Prove that : (i) ABD BAC (ii) BD = AC (iii) ABD BAC 5 In the fig AC = AE and AB = AD and BAD = EAC Prove that BC = DE 6 In fig l m and M is the mid point of the line segment AB Prove that M is also the midpoint of nay line segment CD having its end points on l and m respectively /
9 7 In fig It is given that BC = CE and 1 2 Prove that GCB DCE 8 In fig AD and BC are perpendicular to the line segment AB and AD = BC Prove that O is the mid point of line segment AB and DC 9 In the figure C is the mid point of AB BAD CBE ECA DCB Prove that (i) DAC EBC (ii) DA = EB 1 In the fig BM and DN are both perpendiculars to the segments AC and BM = DN Prove that AC bisects BD 11 In fig, PS = PR, TPS QPR Prove that PT = PQ 12 In fig AD = AE and D and E are points on BC such that BD = EC Prove that AB = AC 13 In the fig AD CD and BC CD If AQ = BP and DP = CQ Prove that DAQ CBP /
10 14 In the fig AB = AC, ABD ACD Prove that BD = CD 15 In fig QPR PQR and M and N are respectively on sides QR and PR of PQR such that QM = PN Prove that OP = OQ, where O is the point of intersection of PM and QN 16 In the fig AB = AC BE and CF are respectively the bisectors of B and C Prove that EBC FCB 17 AD and BE are respectively altitude of ABC such that AE = BD Prove that Ad = BE 18 AD, BE and CF, the altitudes of ABC are equal Prove that ABC is an equilateral triangle 19 AD is the bisector of A of a triangle ABC, P is any point on AD Prove that the perpendicular drawn from P on AB and AC are equal 2 ABCD is a parallelogram, if the two diagonals are equal, find the measure of ABC 21 Fill in the blanks in the following so that each of the following statements is true (i) (ii) (iii) Sides opposite to equal angles of a triangle are Angle opposite to equal sides of a triangle are In an equilateral triangle all angles are (iv) In a ABC if A = C, then AB = (v) (vi) If altitudes CE and BF of a triangle ABC are equal, then AB = In an isosceles triangle ABC with AB = AC, if BD and CE are its altitudes, then BD is CE (vii) In right triangles ABC and DEF, if hypotenuse AB = EF and side AC = DE, then ABC 22 Which of the following statements are true and which are false (i) Sides opposite to equal angles of a triangle are unequal (ii) Angles opposite to equal sides of a triangle are equal (iii) The measure of each angle of an equilateral triangle is /
11 (iv) (v) (vi) If the altitude from one vertex of a triangle bisects the opposite side, then the triangle is isosceles The bisectors of two equal angles of a triangle are equal If the bisector of the vertical angle of a triangle bisects the base, then the triangle may be isosceles (vii) If any two sides of a right triangle are respectively equal to two sides of other right triangle, then the two triangles are congruent (viii) Two right triangles are congruent if hypotenuse and a side of one triangle are respectively equal to the hypotenuse and a side of the other triangle 23 ABC and DBC are two isosceles triangles on the same base BC Show that ABD ACD 24 If ABC is an isosceles triangle with AB = AC Prove that the perpendicular from the vertices B and C to their opposite sides are equal 25 If the altitudes from two vertices of a triangle to the opposite sides are equal Prove that the triangle is isosceles 26 In a right angled triangle, one acute angle is double the other Prove that the hypotenuse is double the smallest side Inequalities in a Triangle (i) (ii) (iii) If two sides of a triangle are unequal, the longer side has greater angle opposite to it If two angles of a triangle are unequal, the greater angle has the longer side opposite to it The sum of any two sides of a triangle is greater than the third side /
12 IX ACADEMIC QUESTIONS Subjective Assignment4 1 Prove that the difference of any two sides of a triangle is less than the third 2 Show that of all the line segments that can be drawn to a given line from a given point not lying on it, the perpendicular line segment is the shortest 3 In a right angled triangle, prove that the hypotenuse is the longest side 4 In the figure, AD is the bisector of A, show that AB > BD 5 In figure PR > PQ and PS bisects QPR Prove that PSR PSQ 6 In the fig PQ > PR QS and RS are the bisector of Q and R respectively Prove that SQ > SR 7 In the fig x y, show that M N /
13 XI SCIENCE & DIP ENTRANCE Subjective Assignment (MCQ) 5 1 In the figure AB CD, 128 o 18 o 2 In the given figure AB CD, ABE 128, (b) 148 o 7 (b) (d) 13 o EFC 3 and 8 1 (d) 13 BED 2, what is the value of EDC? ECF 1, then BAF is equal to 3 In ABC, when BC is produced on both ways, the exterior angles are the value of A 36 (b) (d) and 134, what is 4 In the figure calculate the value of y if 45 (b) 5 6 (d) 9 5 In the figure the value of x is 1 (b) (d) 12 x 5 6 The angles of a triangle are 2x 1, x 2 and x 1 triangle it is? Equilateral Triangle Acute Angle Triangle (b) Right Angled Triangle (d) Obtuse Angle Triangle 7 In the trapezium ABCD, EF AD, what is the value of ACD? 7 (b) 5 (d) 6 4, which type of 8 In the figure CE is perpendicular to AB BDA? 5 (b) 7 (d) 6 9 ACE 2 and ABD 5, what is the measure of /
14 9 In the figure what is the relation of x in term of a, b and c? x a b c (b) x a b c 9 x 18 a b c (d) x 18 a b c 1 PN QR and PM is bisector of P in PQR, then ( Q R ) is equal to MPN (b) 2 MPN 1 MPN (d) 1 MPN The bisector of exterior angles B and C meet at O, what is BOC 12 (b) 8 45 (d) 4 12 The bisectors of exterior angles of ABC intersect at O and form a BOC which is always Acute Angle (b) Right Angle Obtuse Angle (d) None of these 13 The bisectors of interior angles of a triangle forms an angle which is always Acute Angle (b) Right Angle Obtuse Angle (d) All of these 14 In a triangle ABC, the internal bisectors of angles B and C meet at P and the external bisectors of the angles B and C meet at Q, then the BPC BQC is equal to 9 (b) 9 1/ 2 A 9 1/ 2 A (d) In the figure what is the value of a b c? 9 (b) (d) In the figure PM is bisector of P and PN is perpendicular on QR then the value of MPN is 3 (b) 6 (d) BM and CM are interior bisectors of B and C while BN and CN are exterior bisectors of B and C respectively Which is correct? (b) BMC 11 BNC 7 BMC BNC /
15 (d) All are correct 18 In a ABC, AB = 5cm, AC = 5cm and A = 5 o, then B = 35 o (b) 65 o 8 o (d) 4 o 19 If two sides of a triangle are unequal then opposite angle of larger side is greater (b) less equal (d) half 2 The sum of attitudes of a triangle is than the perimeter of the triangle greater (b) less half (d) less 21 In the given figure, PQ = QR, QPR = 48 o, SRP = 18 o, then PQR = 48 o (b) 84 o 3 o (d) 36 o 22 In the given figure, PQR is an equilateral triangle and QRST is a square Then PSR = 3 o (b) 15 o 9 o (d) 6 o 23 Can we draw a triangle ABC with AB = 3cm, BC = 35cm and CA = 65cm? Yes (b) No Can t be determined (d) None of these 24 Which of the following is not a criterion for congruence of triangles? SSA (B) SAS ASA (d) SSS 25 In the given figure, AB BE and EF BE Also BC = DE and AB = EF Then BD = FEC (b) ABD = EFC ABD = CMD (d) ABD = CEF 26 In quadrilateral ABCD, BM and DN are drawn perpendicular to AC such that BM = DN If BR = 8cm, then BD is /
16 4cm (b) 2cm 12cm (d) 16cm 27 In the given figure PQ > PR, QS and RS are the bisectors of Q and R respectively SQ = SR (b) SQ > SR SQ < SR (d) None of these 28 In the figure, PS is the median, bisecting angle P, then QPS is 11 o (b) 7 o 45 o (d) 55 o 29 In the given figure x and y are x = 7 o, y = 37 o (b) x = 37 o, y = 7 o x + y = 117 o (d) x y = 1 o 3 In the given figure BD AC, the measure of ABC is 6 o (b) 3 o 45 o (d) 9 o /
17 ANSWER Assignment 1 2 A = 71 o 5 3 o, 5 o, 1 o 6 52 o, 53 o, 75 o 9 18 o (b) 91 o 4 o, 4 o, 1 o (d) 18 o (e) one (f) one 1 False (b) True False (d) True (e) False (f) True Assignment 2 1, BED = 82 o 3 x = 6 o, y = 55 o 5 x = 115 o, y = 2 o 6 x = 4 o, y = 2 o 7 75 o Assignment (i) equal (ii) equal (iii) 6 (iv) BC (v) AC (vi) Equal to (vii) EFD 22 (i) False (ii) True (iii) True (iv) True (v) True (vi) True (vii) True (viii) True Assignment 5 1c 2d 3b 4b 5d 6b 7a 8b 9c 1d 11c 12a 13c 14d 15d 16a 17d 18b 19a 2b 21b 22b 23b 24a 25a 26d 27b 28c 29b 3d /
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 informationTriangles. 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 informationTriangles. Chapter Flowchart. The Chapter Flowcharts give you the gist of the chapter flow in a single glance.
Triangles Chapter Flowchart The Chapter Flowcharts give you the gist of the chapter flow in a single glance. Triangle A plane figure bounded by three line segments is called a triangle. Types of Triangles
More informationSimilarity 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 informationQuestion 1: In quadrilateral ACBD, AC = AD and AB bisects A (See the given figure). Show that ABC ABD. What can you say about BC and BD?
Class IX  NCERT Maths Exercise (7.1) Question 1: In quadrilateral ACBD, AC = AD and AB bisects A (See the given figure). Show that ABC ABD. What can you say about BC and BD? Solution 1: In ABC and ABD,
More informationClass IX Chapter 7 Triangles Maths. Exercise 7.1 Question 1: In quadrilateral ACBD, AC = AD and AB bisects A (See the given figure).
Exercise 7.1 Question 1: In quadrilateral ACBD, AC = AD and AB bisects A (See the given figure). Show that ABC ABD. What can you say about BC and BD? In ABC and ABD, AC = AD (Given) CAB = DAB (AB bisects
More informationClass IX Chapter 7 Triangles Maths
Class IX Chapter 7 Triangles Maths 1: Exercise 7.1 Question In quadrilateral ACBD, AC = AD and AB bisects A (See the given figure). Show that ABC ABD. What can you say about BC and BD? In ABC and ABD,
More informationCHAPTER 7 TRIANGLES. 7.1 Introduction. 7.2 Congruence of Triangles
CHAPTER 7 TRIANGLES 7.1 Introduction You have studied about triangles and their various properties in your earlier classes. You know that a closed figure formed by three intersecting lines is called a
More informationSOLUTIONS 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 informationCONGRUENCE OF TRIANGLES
Congruence of Triangles 11 CONGRUENCE OF TRIANGLES You might have observed that leaves of different trees have different shapes, but leaves of the same tree have almost the same shape. Although they may
More informationGeometry 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 informationTriangle Congruence and Similarity Review. Show all work for full credit. 5. In the drawing, what is the measure of angle y?
Triangle Congruence and Similarity Review Score Name: Date: Show all work for full credit. 1. In a plane, lines that never meet are called. 5. In the drawing, what is the measure of angle y? A. parallel
More information21. 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 informationTriangles. 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 informationChapter 7. Geometric Inequalities
4. Let m S, then 3 2 m R. Since the angles are supplementary: 3 2580 4568 542 Therefore, m S 42 and m R 38. Part IV 5. Statements Reasons. ABC is not scalene.. Assumption. 2. ABC has at least 2. Definition
More information9 th CBSE Mega Test  II
9 th CBSE Mega Test  II Time: 3 hours Max. Marks: 90 General Instructions All questions are compulsory. The question paper consists of 34 questions divided into four sections A, B, C and D. Section A
More informationHonors Geometry MidTerm Exam Review
Class: Date: Honors Geometry MidTerm 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 informationGeometry 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 informationFill 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 information1. If two angles of a triangle measure 40 and 80, what is the measure of the other angle of the triangle?
1 For all problems, NOTA stands for None of the Above. 1. If two angles of a triangle measure 40 and 80, what is the measure of the other angle of the triangle? (A) 40 (B) 60 (C) 80 (D) Cannot be determined
More informationProperties 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 informationMathematics 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 informationClass 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 informationClass 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 information2. In ABC, the measure of angle B is twice the measure of angle A. Angle C measures three times the measure of angle A. If AC = 26, find AB.
2009 FGCU Mathematics Competition. Geometry Individual Test 1. You want to prove that the perpendicular bisector of the base of an isosceles triangle is also the angle bisector of the vertex. Which postulate/theorem
More informationRevision Question Bank
Revision Question Bank Triangles 1. In the given figure, find the values of x and y. Since, AB = AC C = B [angles opposite to the equal sides are equal] x = 50 Also, the sum of all angles of a triangle
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
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 informationSSC 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 informationAREAS OF PARALLELOGRAMS AND TRIANGLES
AREAS OF PARALLELOGRAMS AND TRIANGLES Main Concepts and Results: The area of a closed plane figure is the measure of the region inside the figure: Fig.1 The shaded parts (Fig.1) represent the regions whose
More informationPRACTICE 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 informationLLT 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 informationClass 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 informationGeometry. Midterm Review
Geometry Midterm Review Class: Date: Geometry Midterm Review Multiple Choice Identify the choice that best completes the statement or answers the question. 1 A plumber knows that if you shut off the water
More informationExercise 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 informationEXERCISE 10.1 EXERCISE 10.2
NCERT Class 9 Solved Questions for Chapter: Circle 10 NCERT 10 Class CIRCLES 9 Solved Questions for Chapter: Circle EXERCISE 10.1 Q.1. Fill in the blanks : (i) The centre of a circle lies in of the circle.
More informationChapter (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 informationHonors 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 information0609ge. Geometry Regents Exam AB DE, A D, and B E.
0609ge 1 Juliann plans on drawing ABC, where the measure of A can range from 50 to 60 and the measure of B can range from 90 to 100. Given these conditions, what is the correct range of measures possible
More informationKENDRIYA VIDYALAYA SANGATHAN, HYDERABAD REGION
KENDRIYA VIDYALAYA SANGATHAN, HYDERABAD REGION SAMPLE PAPER 02 F PERIODIC TEST III EXAM (201718) SUBJECT: MATHEMATICS(041) BLUE PRINT : CLASS IX Unit Chapter VSA (1 mark) SA I (2 marks) SA II (3 marks)
More informationQUESTION 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 informationChapter  7. (Triangles) Triangle  A closed figure formed by three intersecting lines is called a triangle. A
Chapter  7 (Triangles) Triangle  A closed figure formed by three intersecting lines is called a triangle. A triangle has three sides, three angles and three vertices. Congruent figures  Congruent means
More informationIB MYP Unit 6 Review
Name: Date: 1. Two triangles are congruent if 1. A. corresponding angles are congruent B. corresponding sides and corresponding angles are congruent C. the angles in each triangle have a sum of 180 D.
More informationDownloaded from
Triangles 1.In ABC right angled at C, AD is median. Then AB 2 = AC 2  AD 2 AD 2  AC 2 3AC 24AD 2 (D) 4AD 23AC 2 2.Which of the following statement is true? Any two right triangles are similar
More information1 What is the solution of the system of equations graphed below? y = 2x + 1
1 What is the solution of the system of equations graphed below? y = 2x + 1 3 As shown in the diagram below, when hexagon ABCDEF is reflected over line m, the image is hexagon A'B'C'D'E'F'. y = x 2 + 2x
More informationGeometry Honors: Midterm Exam Review January 2018
Name: Period: The midterm will cover Chapters 16. Geometry Honors: Midterm Exam Review January 2018 You WILL NOT receive a formula sheet, but you need to know the following formulas Make sure you memorize
More information0113ge. 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 informationTopic 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 informationNozha 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) 1Complete 1. in the parallelogram, each two opposite
More informationMathematics 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 information10. 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 noncollinear points. Mark (1) Q 2 State the following statement as true or false. Give reasons also.the perpendicular
More informationQ.2 A, B and C are points in the xy plane such that A(1, 2) ; B (5, 6) and AC = 3BC. Then. (C) 1 1 or
STRAIGHT LINE [STRAIGHT OBJECTIVE TYPE] Q. A variable rectangle PQRS has its sides parallel to fied directions. Q and S lie respectivel on the lines = a, = a and P lies on the ais. Then the locus of R
More informationMOCKTIME.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 informationGeometry 21  More Midterm Practice
Class: Date: Geometry 21  More Midterm Practice 1. What are the names of three planes that contain point A? 6. If T is the midpoint of SU, what are ST, TU, and SU? A. ST = 7, TU = 63, and SU = 126 B.
More informationYear 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 informationReview for Geometry Midterm 2015: Chapters 15
Name Period Review for Geometry Midterm 2015: Chapters 15 Short Answer 1. What is the length of AC? 2. Tell whether a triangle can have sides with lengths 1, 2, and 3. 3. Danny and Dana start hiking from
More informationNozha Directorate of Education Form : 2 nd Prep
Cairo Governorate Department : Maths Nozha Directorate of Education Form : 2 nd Prep Nozha Language Schools Geometry Revision Sheet Ismailia Road Branch Sheet ( 1) 1Complete 1. In the parallelogram, each
More information1 Line n intersects lines l and m, forming the angles shown in the diagram below. 4 In the diagram below, MATH is a rhombus with diagonals AH and MT.
1 Line n intersects lines l and m, forming the angles shown in the diagram below. 4 In the diagram below, MATH is a rhombus with diagonals AH and MT. Which value of x would prove l m? 1) 2.5 2) 4.5 3)
More informationVAISHALI 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 information0112ge. Geometry Regents Exam Line n intersects lines l and m, forming the angles shown in the diagram below.
Geometry Regents Exam 011 011ge 1 Line n intersects lines l and m, forming the angles shown in the diagram below. 4 In the diagram below, MATH is a rhombus with diagonals AH and MT. Which value of x would
More informationChapter 3 Cumulative Review Answers
Chapter 3 Cumulative Review Answers 1a. The triangle inequality is violated. 1b. The sum of the angles is not 180º. 1c. Two angles are equal, but the sides opposite those angles are not equal. 1d. The
More informationGEO REVIEW TEST #1. 1. In which quadrilateral are the diagonals always congruent?
GEO REVIEW TEST #1 Name: Date: 1. In which quadrilateral are the diagonals always congruent? (1) rectangle (3) rhombus 4. In the accompanying diagram, lines AB and CD intersect at point E. If m AED = (x+10)
More informationQ.1 If a, b, c are distinct positive real in H.P., then the value of the expression, (A) 1 (B) 2 (C) 3 (D) 4. (A) 2 (B) 5/2 (C) 3 (D) none of these
Q. If a, b, c are distinct positive real in H.P., then the value of the expression, b a b c + is equal to b a b c () (C) (D) 4 Q. In a triangle BC, (b + c) = a bc where is the circumradius of the triangle.
More informationA 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 informationVisit: ImperialStudy.com For More Study Materials Class IX Chapter 12 Heron s Formula Maths
Exercise 1.1 1. Find the area of a triangle whose sides are respectively 150 cm, 10 cm and 00 cm. The triangle whose sides are a = 150 cm b = 10 cm c = 00 cm The area of a triangle = s(s a)(s b)(s c) Here
More informationHonors Geometry Term 1 Practice Final
Name: Class: Date: ID: A Honors Geometry Term 1 Practice Final Short Answer 1. RT has endpoints R Ê Ë Á 4,2 ˆ, T Ê ËÁ 8, 3 ˆ. Find the coordinates of the midpoint, S, of RT. 5. Line p 1 has equation y
More information0809ge. 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 informationCHAPTER 10 CIRCLES Introduction
168 MATHEMATICS CIRCLES CHAPTER 10 10.1 Introduction You may have come across many objects in daily life, which are round in shape, such as wheels of a vehicle, bangles, dials of many clocks, coins of
More informationChapter 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 informationMathematics. Exercise 6.4. (Chapter 6) (Triangles) (Class X) Question 1: Let and their areas be, respectively, 64 cm 2 and 121 cm 2.
() Exercise 6.4 Question 1: Let and their areas be, respectively, 64 cm 2 and 121 cm 2. If EF = 15.4 cm, find BC. Answer 1: 1 () Question 2: Diagonals of a trapezium ABCD with AB DC intersect each other
More information0610ge. 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 informationGeometry Midterm Exam Review 3. Square BERT is transformed to create the image B E R T, as shown.
1. Reflect FOXY across line y = x. 3. Square BERT is transformed to create the image B E R T, as shown. 2. Parallelogram SHAQ is shown. Point E is the midpoint of segment SH. Point F is the midpoint of
More information8. 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 informationLabel 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 informationExercises for Unit I I I (Basic Euclidean concepts and theorems)
Exercises for Unit I I I (Basic Euclidean concepts and theorems) Default assumption: All points, etc. are assumed to lie in R 2 or R 3. I I I. : Perpendicular lines and planes Supplementary background
More informationCalgary Math Circles: Triangles, Concurrency and Quadrilaterals 1
Calgary Math Circles: Triangles, Concurrency and Quadrilaterals 1 1 Triangles: Basics This section will cover all the basic properties you need to know about triangles and the important points of a triangle.
More informationGeometry  Review for Final Chapters 5 and 6
Class: Date: Geometry  Review for Final Chapters 5 and 6 1. Classify PQR by its sides. Then determine whether it is a right triangle. a. scalene ; right c. scalene ; not right b. isoceles ; not right
More informationBOARD 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 information1. In a triangle ABC altitude from C to AB is CF= 8 units and AB has length 6 units. If M and P are midpoints of AF and BC. Find the length of PM.
1. In a triangle ABC altitude from C to AB is CF= 8 units and AB has length 6 units. If M and P are midpoints of AF and BC. Find the length of PM. 2. Let ABCD be a cyclic quadrilateral inscribed in a circle
More information0611ge. Geometry Regents Exam Line segment AB is shown in the diagram below.
0611ge 1 Line segment AB is shown in the diagram below. In the diagram below, A B C is a transformation of ABC, and A B C is a transformation of A B C. Which two sets of construction marks, labeled I,
More information86. 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 86. a: 110 b: 70 c: 48 d: 108 87. a: no b: yes c: no d: yes e: no f: yes g: yes h: no 88. b: The measure of an exterior angle of a triangle equals the sum of the measures of its remote
More information0612ge. Geometry Regents Exam
0612ge 1 Triangle ABC is graphed on the set of axes below. 3 As shown in the diagram below, EF intersects planes P, Q, and R. Which transformation produces an image that is similar to, but not congruent
More informationKENDRIYA VIDYALAYA SANGATHAN, HYDERABAD REGION
KENDRIYA VIDYALAYA SANGATHAN, HYDERABAD REGION SAMPLE PAPER 02 F SESSING ENDING EXAM (201718) SUBJECT: MATHEMATICS(041) BLUE PRINT : CLASS IX Unit Chapter VSA (1 mark) SA I (2 marks) SA II (3 marks) LA
More information0110ge. 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 information86. 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 86. a: 110 b: 70 c: 48 d: 108 87. a: no b: yes c: no d: yes e: no f: yes g: yes h: no 88. b: The measure of an exterior angle of a triangle equals the sum of the measures of its remote
More informationMathematics. A basic Course for beginers in G.C.E. (Advanced Level) Mathematics
Mathematics A basic Course for beginers in G.C.E. (Advanced Level) Mathematics Department of Mathematics Faculty of Science and Technology National Institute of Education Maharagama Sri Lanka 2009 Director
More information0811ge. 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 informationTHEOREMS WE KNOW PROJECT
1 This is a list of all of the theorems that you know and that will be helpful when working on proofs for the rest of the unit. In the Notes section I would like you to write anything that will help you
More informationUNIT8 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 rightangled triangle, rightangled
More informationHigher 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 informationGeometry  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 informationAnswer Key. 9.1 Parts of Circles. Chapter 9 Circles. CK12 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 informationGeometry Practice Midterm
Class: Date: Geometry Practice Midterm 201819 1. If Z is the midpoint of RT, what are x, RZ, and RT? A. x = 19, RZ = 38, and RT = 76 C. x = 17, RZ = 76, and RT = 38 B. x = 17, RZ = 38, and RT = 76 D.
More informationName: 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 informationQuestion 1 ( 1.0 marks) places of decimals? Solution: Now, on dividing by 2, we obtain =
Question 1 ( 1.0 marks) The decimal expansion of the rational number places of decimals? will terminate after how many The given expression i.e., can be rewritten as Now, on dividing 0.043 by 2, we obtain
More informationHigher 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 informationThe 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 informationSHW 101 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 informationright angle an angle whose measure is exactly 90ᴼ
right angle an angle whose measure is exactly 90ᴼ m B = 90ᴼ B two angles that share a common ray A D C B Vertical Angles A D C B E two angles that are opposite of each other and share a common vertex two
More informationCBSE Class IX Mathematics Term 1. Time: 3 hours Total Marks: 90. Section A
CBSE sample papers, Question papers, Notes for Class 6 to 1 CBSE Class IX Mathematics Term 1 Time: 3 hours Total Marks: 90 General Instructions: 1. All questions are compulsory.. The question paper consists
More informationGeometry Advanced Fall Semester Exam Review Packet  CHAPTER 1
Name: Class: Date: Geometry Advanced Fall Semester Exam Review Packet  CHAPTER 1 Multiple Choice. Identify the choice that best completes the statement or answers the question. 1. Which statement(s)
More information