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

Size: px
Start display at page:

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

Transcription

1 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!!! " " + DA + AB + BC f)!!! "!!! "!!! "!!! " AB + AD + BA + DC AB + BA + AD + DC AC. Consider the parallelogram shown alongside. Which of the following statements are true? a) AB DC : true b) a b : false c) BC b : true d) AC + CD b : true e) AD CB : false. A triangle ABC is given by AB c and BC a. Point D is the midpoint of the side AB. Express the vectors AC, AD and CD in terms of a and c. " " AC a + c AD c" CD a " + c". In a parallelogram ABCD, let AB a and BC b. Point E is the midpoint of AB ; point F lies on BC so that BF : FC : holds. Express the vectors AE, AC, BD, CD, DE, BF, AF and EF in terms of a and b. AE a" " " AC a + b " " BD b a CD a " DE a" b " BF! "!! b"

2 chapter vector geometry solutions V. " AF a + b" EF a" + b". In a plane quadrilateral ABCD, let AB a, BC b and CD c. a) Express d DA in terms of a, b and c. d! a! + b! + c! ( ) b) Use vectors to show that the midpoints of the sides of the quadrilateral are vertices of a parallelogram. EF a" + b " ( ) HG d" + c " #!!! " CA a" b " Thus: EF HG ( ) ( a" + b " ). Let S be the centre of gravity of a triangle ABC. Prove: SA + SB + SC. SA! MA b" + a" SB c" + b" SC!!! " a" + c" SA + SB + SC!!! " a " + b " + c " " 7. Prove: A quadrilateral whose diagonals bisect each other is a parallelogram. a! + f "! e!! c! f "! + e!! a! + c!! a! c!

3 chapter vector geometry solutions V. Exercise B. Given a! a a. How do you find the magnitude (length) a! of the vector a!? a! a + a (Pythagoras). Given a a a and b b b. How do you calculate a) c a + b? c! a! + b! a a + b b a e! + a e! ( ) + ( b e! + b e! ) ( a + b ) e! + ( a + b ) e! a + b a + b b) c a b? c! a! b! a a b b a e! + a e! ( ) ( b e! + b e! ) ( a b ) e! + ( a b ) e! a b a b c) c k a? c! k a! k a a ( ) k a e! + a e! ( k a ) e! + ( k a ) e! k a k a

4 chapter vector geometry solutions V.!!! ".. Given A( x A / y A ) and B( x B / y B ). Find AB!!! " x b x a AB OA + OB OB OA y b y a (head minus tail). Given a and b. How do you check (without drawing the vectors) if a and b are parallel (or antiparallel) to each other? b! k a! k > a! and b! are parallel k < a! and b! are antiparallel. Let a, b, c. The vectors d a b + c and e x are collinear. Find x. x ( k )

5 chapter vector geometry solutions V.. Complete the parallelogram ABCD where A, B and C are given. What are the coordinates of D? ( ), B( 8 /), C( / ) a) A / OD OA + BC D( /) + b) A( / ), B( 7 / ), C( / 7) D( /8) 7. Express vector c in terms of vectors a and b ( c k a + k b ). a) a c! a! + b! b) a c! a! b! ; b ; b 8 ; c ; c Given A( 9 / ), B( / ), C( /). Calculate the perimeter of the triangle ABC. perimeter: 8 9. Given A( / ), B( 7 / ). Find the coordinates of the midpoint of AB.!!!!!! "! OM AB OA + AB OA + OB OA OA + OB M AB ( /) ( ) ( )

6 chapter vector geometry solutions V.. Given A( / ), B( / 7), C( / ). Find the coordinates of the centre of gravity of the triangle ABC.! OS OA + AB! + M!!!!! " ABC OA + OB ( OA ) + ( OC OM!!!! " AB ) OA + OB ( OA ) + OC OA ( + OB ) OA OA OA + OB OB + OC OA + OB + OC S( / ) ( ) Exercise C. Given a a a a. How do you find the magnitude (length) a of the vector a? a! a a a a + a + a. Determine whether vectors a and b are collinear. a) a a!, b 9 and b! are not collinear, because ( ) and ( ) ( ) but 9

7 chapter vector geometry solutions V. b) a 7, b.. a! and b! are collinear, because b! a!. Determine whether point C lies on the line passing through A and B. a) A( / / ) ; B( / / ) ; C( 8 / 8 / ) yes, because! AC!!! " AB!!! " AB ;! AC b) A( / / ) ; B( / 7 / ) ; C( / / 8) no, because there is no k! so that! AC!!! " k AB!!! " AB ;! AC 9. Determine whether A( / / ), B( / / ) and C ( / 9 / ) are vertices of a triangle ABC. You have to show that A, B and C are on the same line or not.!!! " AB ;! AC 9 9! AC!!! " AB A, B and C are on the same line and therefore A, B and C are not the vertices of a triangle.. The points A( / / ), B( / / ) and C ( / / 7) are vertices of a parallelogram. Determine the coordinates of the fourth vertex D (three results!). D ( / / ); D ( / / ); D / 9 / ( ). Calculate the length of vector a direction and length... Find the components of a vector with the same a! 9; 7. 7

8 chapter vector geometry solutions V. 7. Calculate the length of vector a 9. Find the components of a vector of opposite direction and length of. a! ; Find the points on the z -axis where the distance to P( / / 7) is 7. ( ) PQ Q / / z PQ 7 z ;z 9 z 7 Q ( / / );Q ( / / 9) Exercise D. Find the angle between a and b where the following is satisfied: a) a b a b α b) a! b! a! b! α. Calculate the scalar product a b. a) a!, b! a! b! b) a!, b! a! b! 8

9 chapter vector geometry solutions V.. Find the angle between the vectors a and b. a) a, b α 9.9 b) a, b 7 α 8.. Calculate the angle between a and the coordinate axes. a) a α.87;α.8 b) a α 9. ;α.88 ;α 77.. Find the possible values of u given that a and b are perpendicular. a) a 7, b u u b) a u u 7, b u + 9 u + u ;u. Determine the point P given that APB 9. a) A( / / 8), B( / / ), P on the y -axis. P ( / / );P ( / / ) b) A( / /), B( / / ), P on the x -axis. P ( / / );P ( / / ) 9

10 chapter vector geometry solutions V. 7. The vectors a x and b z form sides of a square. Calculate the area of this square. A 9 with x and z A. with x z. Exercise E. For each of the following determine the vector and the Cartesian equation. The line passes a) through A( / 7) and its gradient m equals to ; vector equation: r! 7 + t Cartesian equation: x y b) through A( /) and B( / ) ; vector equation: r! + t 7 Cartesian equation: x 7y + c) through A( / ) and intersects the y -axis at y ; vector equation: r! + t Cartesian equation: x + y d) through A( / 7) and is parallel to the y -axis; vector equation: r! 7 + t Cartesian equation: x + e) through A( 8 / ) and is parallel to the x -axis. vector equation: r! 8 + t Cartesian equation: y +

11 chapter vector geometry solutions V.. Calculate the point and angle of intersection of the lines l and l. a) l :r + s ; l :r 9 + t S( / 7);ϕ 7.9 b) l :r 8 + s ; l :r + t l l c) l : y x 7 ; l : y 7x S( / );ϕ.9 d) l :x y ; l : x + y S( / );ϕ 9. Show that the lines l :r! + s and l :r! + t are normal. with the dot product of the direction vectors: Exercise F. Find a vector equation for the line that passes through a) A( / / ) and B( / / ) ;! 7 r + t b) A( / / ) and intersects the x -axis at x ;! r + t c) A( / / ) and is parallel to the z -axis;! r + t

12 chapter vector geometry solutions V. d) A( 7 / / ) and is parallel to the y -axis.! 7 r + t. Given the points A( // ) and B( / / ), find a vector equation of the line that passes through the midpoint of the segment AB and is parallel to the x -axis. l :r!. + t. Are the points A( / / ) and B( / / 7) on the line l :r A yes, B no + t?. The segment with end-points A( / / ) and B( / / ) is divided into three equal parts. Find the coordinates of the dividing points. P ( / / ) ; P ( // ). Are the following lines skew, parallel, coincident or intersecting? a) l :r! + s, l :r! + t skew b) l :r.8 + s.., l :r + t parallel c) l :r. + s.., l :r + t identical d) l :r + s, l :r intersecting at S( / / 7) + t

13 chapter vector geometry solutions V. e) l :r 7 + s, l :r + t. parallel. Find the point of intersection S and the acute angle between l :r + s and l :r + s 7. S( / 8 /) ;. 7. In a triangle ABC the vertices A( / / ) and B( 7 / 9 / ) are given. The vertex C lies on the line through P( // ) and Q( // ). Calculate the coordinates of the vertex C given that the side c AB a) is the hypotenuse of the right-angled triangle ABC, C ( // ) ; C ( // 8) b) is the base of the isosceles triangle ABC. C // Exercise G. Determine the Cartesian equation of the plane ABC. a) A( // 7), B( / / ), C( / / ) x + y z b) A( / / ), B( 7 / / ), C ( / / ) x y + z. Determine the Cartesian equation of the following planes given by ( ) and l :r! a) P / / y z t b) A( / /) and B( / / ) ; also, the z -axis is parallel to the plane. x 7y

14 chapter vector geometry solutions V.. Determine the Cartesian equation of the plane containing a) l :r! + s and l :r! + t ; x b) P( /. /), parallel to the xz -plane; y + c) l :r! 8 + s and l :r! 8 + t. z 8. Describe the particular position of the following planes. a) ε : x 7y + ε z axis b) ε :y z + 9 ε x axis c) ε : x ε yz plane d) ε :x + z ε through y axis. Prove that the lines l and l intersect; determine the Cartesian equation of the plane containing l and l. a) l :r! + s 7, l :r! x y + z ;S( / / ) + t b) l :r! 8 + s, l :r! 8x y z ;S( // ) 8 + t

15 chapter vector geometry solutions V.. Calculate the axes intercepts of the plane ε. a) ε : x y + z a ;b ;c b) ε : x z + a ;c 7. Determine the Cartesian equation for the plane with intercepts x a, y b and z c. Then divide the equation by abc. bcx + acy + abz abc resp. x a + y b + z (axes intercept form for the equations for planes) c 8. A plane is given by its axes intercepts. Determine its Cartesian equation. a) a, b, c x + y + z b) a 8, b, c x y + z c) a, b 7 7x + y + d) a 7 x 7 9. Determine vector equations of the trace lines of plane ε. The trace lines of a plane ε are the lines of intersection between the plane ε and the xy -, xz - respectively yz -plane. a) ε : x y + z l : x y z + t l : x y z + t l : x y z + t

16 chapter vector geometry solutions V. b) ε : y z + 8 l : x y z + t l : x y z + t l : x y z 8 + t. Find point A in plane ε :r! 9 + s 8 + t a) which lies on the z -axis, A( / / ) b) which possesses three equal components, A(. /. /.) c) with horizontal projection A' ( / / ). A( / / ). Determine whether A( //), B( / / ), C( / / ) and D( / / ) are vertices of a quadrilateral. yes ( ε :8x 9y z + 8 ). Find the point at which the line l intersects the plane ε. a) l :r! 8 P( / / ) + t, ε : x y + z b) l :r l ε t, ε : y z + 7

17 chapter vector geometry solutions V. c) l :r + t 7, ε :r! 7 + s + t 8 P( / / ) d) l :r 9 + t, ε :r! 9 + s + t l ε. Determine a vector equation of the line of intersection between planes ε and ε. a) ε : x y + z, ε : x + y z + l : x y z + t b) ε : x + y z +, ε : x y + z l : x y z + t Exercise H. Show in different ways that A( 7 / /), B( 9 / / 7) and C( / 7 / ) lie on the same line.!!! " AB ;! AC 8 I)! AC! AB!!! " AB and AC are collinear II) III)!!! " " AB AC!!! "!!! " AB AC AB AC or!!! "!!! " AB AC AB AC!!! "!!! " here: AB AC AB AC 7

18 chapter vector geometry solutions V.. A line l passes through P / / ( ) and is perpendicular to the lines r + s and r + t. Find the vector equation of l. l :r! + t + t 7 7. Find a vector that is perpendicular to the plane containing the points A( / /), B( / / ) and C ( / / ).!!! " AB ;! AC v! """! """! 9 AB AC or v! 9. Calculate the area of the triangle A( 7 / /), B( / / ), C( 9 / /).. Find the area of the parallelogram having the diagonals e! "! e f and f. Exercise I. Determine the Cartesian equation of the plane that passes through point P and is parallel to the plane ε. a) P( / / ) ; ε :x y + z + x y + z b) P( // ) ; ε : x + z x + z 8

19 chapter vector geometry solutions V.. Determine the Cartesian equation of the plane that contains the point P and is normal to the line l. ( ) ; l :r! a) P / / x + y z + + t ( ) ; l :r b) P / / x + y z 8 + t. Which point on the line l : r + t is equidistant from the points A ( / / ) and B( / / 7)? P( // ). Determine the Cartesian equation of the plane which contains the points P( / / ) and Q( / /) and which is normal to the plane ε : x + y. x y z + 9. Point P( / / ) is reflected in the plane ε : x y + z +. Find the coordinates of the image point P'. P' ( // ). Point P' ( / / 7) is the reflection point of P( / / ). Find the Cartesian equation for the plane ε in which P was reflected. x + y 9z + 7. The line l : r 7 + t is reflected in the plane ε : x + y z +. Determine a vector equation of the reflected line l '. l ' :r! + t 9

20 chapter vector geometry solutions V. 8. A ray of light passes through P( 7 / 7 / ) and is reflected in the plane ε : x y + z. Point Q( 7 / / 8) lies on the reflected ray of light. At which point in the plane ε is the ray of light reflected? R( / / ) 9. Determine the acute angle between the planes ε : x y z + and ε : x + z. α.. Determine the angle of intersection of line l : r + t and plane ε : x y +. α 7. Exercise J. What is the distance between point P( / ) and the line l : x + y? d. Find the length of the perpendicular to the line l : r 9 + t from the origin. d. Calculate the distance between point P( / /) and plane ε : x + 7y z +. d. Given the plane ε : x y 7z +. Find another plane ε passing through point P( / /) and parallel to ε. Determine the Cartesian equation of the plane ε as well as the distance between the two planes. ε : x y 7z, d. Find the distance between the skew lines l and l. a) l :r! + t, l :r! + t d b) l : r + t, l : r + t d

21 chapter vector geometry solutions V.. Determine the Cartesian equation of the plane that is parallel to the plane ε and d units away from it. a) ε :x y +z, d ε :x y +z + and ε :x y +z b) ε : x 7z +, d ε :x 7z + and ε :x 7z 9 7. Determine the Cartesian equation of the angle bisector planes of the planes ε and ε. a) ε : x y z +, ε : x + y z + β :x + y 8z +, β :x y 7 b) ε : x + y + 7z, ε :x y z + 9 β :x y +z + 7, β :9x y z + 8. Which points of the line l : r + t are equidistant from the planes ε : x y + and ε : x + z + 7? P ( / / ), P ( / / ) Exercise K. Determine the centre O and the radius r of the circle c. a) c : x + y 8x + y O( / ) ; r b) c : x + y + x 8y + 8 O / ; r. Determine the points of intersection of the circle c : x + y and the line l : y x +. P ( /) ; P ( 8 / ). Point A( / ) lies on a circle with a radius r. The centre of the circle lies on the line l : y x +. Determine the equation of the circle. c :( x ) + ( y ) ; c :( x +) + ( y )

22 chapter vector geometry solutions V.. Determine the equation of the circle with the centre O( / ) which touches the line l : x y +. ( x ) + ( y ). Find the equation of the circumcircle of the triangle A( / ), B( 8 / ) and C( / ). ( x ) + ( y ). The circle c :( x ) + ( y + ) 9 is touched by circles, each with radius r, whose centres lie on the line l : x 7y +. Find the equations of the circles. c :( x 9) + ( y ) 9 ; c :( x + ) + ( y ) 9 7. The circles c : x + y 9 and c :( x ) + ( y ) '9 have a common chord. What is its length? P (. /.) ; P (. /.) ; P P 8. A circle passes through A( /) and B( / 7) and touches l : x. Find its equation. c :( x ) + ( y + ) 9 ; c :( x + 9) + ( y 8) '89 9. Find the Cartesian equation of the tangent at the point P( / 9) of the circle c :( x ) + ( y + ) 9. x y +. Find the equations for the tangents of the circle c : x + y through the point P( / ). t : x y + ; t : x + y +. Determine the centre O and the radius r of the sphere S. a) S : x + y + z x + y z + O( / / ) ; r b) S : x + y + z + x z + 9 O( / / ) ; r. Find the points of intersection between the sphere S :( x ) + ( y + ) + z 8 and the line l passing through A( 9 / /) and B( / 7 / ). P ( // ) ; P ( / /)

23 chapter vector geometry solutions V.. The spheres S :( x ) + ( y + ) + z and S : x + y + z x y + z + are given. a) Show that S and S touch each other. O O r + r b) Determine the point of contact. P( 7 / / ). Find the equation of the sphere with centre O( / / ) which touches the plane ε : x y + z +. ( x ) + ( y ) + ( z + ). Find the equations of the spheres with centre O( 9 / / ) which touch the sphere S :( x ) + ( y + ) + ( z ) 9. S :( x 9) + ( y ) + ( z ) ; S :( x 9) + ( y ) + ( z ). Point P( / / ) lies on a sphere with radius r. The centre of the sphere lies on the line that passes through A( / / ) and B( / 7 / ). Find the equation of the sphere. S :( x ) + ( y ) + ( z + ) 9 ; S :( x ) + ( y ) + ( z + ) 9 7. Point P( / / ) lies on the sphere with the centre O( / / ) and the radius r. Find the equation of the tangent plane to the sphere at point P. y z 8. Determine the Cartesian equations for the tangent planes to the sphere S :( x ) + ( y ) + ( z + ) 9 which are parallel to the plane ε : x + y z. τ : x + y z 7 ; τ : x + y z + 9. A ray of light, starting at the light source Q( / 8 / 7), travels in the direction of P( / / ) and is reflected in the sphere S :( x ) + ( y + 8) + z. a) Find the point R on the sphere where the reflection takes place. R( / / ) b) Determine a vector equation of the reflected ray of light.! r + t

24 chapter vector geometry solutions V. c) Find the angle between the rays at point R..

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,

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.

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

Sample Question Paper Mathematics First Term (SA - I) Class IX. Time: 3 to 3 ½ hours

Sample Question Paper Mathematics First Term (SA - I) Class IX Time: 3 to 3 ½ hours M.M.:90 General Instructions (i) All questions are compulsory. (ii) The question paper consists of 34 questions divided

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,

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?

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

Unit 8. ANALYTIC GEOMETRY.

Unit 8. ANALYTIC GEOMETRY. 1. VECTORS IN THE PLANE A vector is a line segment running from point A (tail) to point B (head). 1.1 DIRECTION OF A VECTOR The direction of a vector is the direction of the

2D VECTORS Question 1 (**) Relative to a fixed origin O, the point A has coordinates ( 2, 3). The point B is such so that AB = 3i 7j, where i and j are mutually perpendicular unit vectors lying on the

8. Find r a! r b. a) r a = [3, 2, 7], r b = [ 1, 4, 5] b) r a = [ 5, 6, 7], r b = [2, 7, 4]

Chapter 8 Prerequisite Skills BLM 8-1.. Linear Relations 1. Make a table of values and graph each linear function a) y = 2x b) y = x + 5 c) 2x + 6y = 12 d) x + 7y = 21 2. Find the x- and y-intercepts of

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

POINT. Preface. The concept of Point is very important for the study of coordinate

POINT Preface The concept of Point is ver important for the stud of coordinate geometr. This chapter deals with various forms of representing a Point and several associated properties. The concept of coordinates

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

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

Additional Mathematics Lines and circles Topic assessment 1 The points A and B have coordinates ( ) and (4 respectively. Calculate (i) The gradient of the line AB [1] The length of the line AB [] (iii)

DATE: MATH ANALYSIS 2 CHAPTER 12: VECTORS & DETERMINANTS

NAME: PERIOD: DATE: MATH ANALYSIS 2 MR. MELLINA CHAPTER 12: VECTORS & DETERMINANTS Sections: v 12.1 Geometric Representation of Vectors v 12.2 Algebraic Representation of Vectors v 12.3 Vector and Parametric

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)

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

b UVW is a right-angled triangle, therefore VW is the diameter of the circle. Centre of circle = Midpoint of VW = (8 2) + ( 2 6) = 100

Circles 6F a U(, 8), V(7, 7) and W(, ) UV = ( x x ) ( y y ) = (7 ) (7 8) = 8 VW = ( 7) ( 7) = 64 UW = ( ) ( 8) = 8 Use Pythagoras' theorem to show UV UW = VW 8 8 = 64 = VW Therefore, UVW is a right-angled

Worksheet A VECTORS 1 G H I D E F A B C

Worksheet A G H I D E F A B C The diagram shows three sets of equally-spaced parallel lines. Given that AC = p that AD = q, express the following vectors in terms of p q. a CA b AG c AB d DF e HE f AF

(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

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

arxiv: v1 [math.ho] 29 Nov 2017

The Two Incenters of the Arbitrary Convex Quadrilateral Nikolaos Dergiades and Dimitris M. Christodoulou ABSTRACT arxiv:1712.02207v1 [math.ho] 29 Nov 2017 For an arbitrary convex quadrilateral ABCD with

COORDINATE GEOMETRY BASIC CONCEPTS AND FORMULAE. To find the length of a line segment joining two points A(x 1, y 1 ) and B(x 2, y 2 ), use

COORDINATE GEOMETRY BASIC CONCEPTS AND FORMULAE I. Length of a Line Segment: The distance between two points A ( x1, 1 ) B ( x, ) is given b A B = ( x x1) ( 1) To find the length of a line segment joining

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

Alg. (( Sheet 1 )) [1] Complete : 1) =.. 3) =. 4) 3 a 3 =.. 5) X 3 = 64 then X =. 6) 3 X 6 =... 7) 3

Cairo Governorate Department : Maths Nozha Directorate of Education Form : 2 nd Prep. Nozha Language Schools Sheet Ismailia Road Branch [1] Complete : 1) 3 216 =.. Alg. (( Sheet 1 )) 1 8 2) 3 ( ) 2 =..

0611ge. 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,

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

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

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 =,

Nozha 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) 1-Complete 1. In the parallelogram, each

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

2. 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

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

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

CBSE CLASS X MATH -SOLUTION Therefore, 0.6, 0.25 and 0.3 are greater than or equal to 0 and less than or equal to 1.

CBSE CLASS X MATH -SOLUTION 011 Q1 The probability of an event is always greater than or equal to zero and less than or equal to one. Here, 3 5 = 0.6 5% = 5 100 = 0.5 Therefore, 0.6, 0.5 and 0.3 are greater

KENDRIYA VIDYALAYA SANGATHAN, HYDERABAD REGION SAMPLE PAPER 01 F PERIODIC TEST III EXAM (2017-18) SUBJECT: MATHEMATICS(041) BLUE PRINT : CLASS IX Unit Chapter VSA (1 mark) SA I (2 marks) SA II (3 marks)

Visit: 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

[BIT Ranchi 1992] (a) 2 (b) 3 (c) 4 (d) 5. (d) None of these. then the direction cosine of AB along y-axis is [MNR 1989]

VECTOR ALGEBRA o. Let a i be a vector which makes an angle of 0 with a unit vector b. Then the unit vector ( a b) is [MP PET 99]. The perimeter of the triangle whose vertices have the position vectors

1. The unit vector perpendicular to both the lines. Ans:, (2)

1. The unit vector perpendicular to both the lines x 1 y 2 z 1 x 2 y 2 z 3 and 3 1 2 1 2 3 i 7j 7k i 7j 5k 99 5 3 1) 2) i 7j 5k 7i 7j k 3) 4) 5 3 99 i 7j 5k Ans:, (2) 5 3 is Solution: Consider i j k a

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

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 Undefined Terms: Point, Line, Incident, Between, Congruent. Incidence Axioms:

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

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

Vectors - Applications to Problem Solving

BERKELEY MATH CIRCLE 00-003 Vectors - Applications to Problem Solving Zvezdelina Stankova Mills College& UC Berkeley 1. Well-known Facts (1) Let A 1 and B 1 be the midpoints of the sides BC and AC of ABC.

INVERSION IN THE PLANE BERKELEY MATH CIRCLE

INVERSION IN THE PLANE BERKELEY MATH CIRCLE ZVEZDELINA STANKOVA MILLS COLLEGE/UC BERKELEY SEPTEMBER 26TH 2004 Contents 1. Definition of Inversion in the Plane 1 Properties of Inversion 2 Problems 2 2.

KENDRIYA VIDYALAYA SANGATHAN, HYDERABAD REGION SAMPLE PAPER 02 F PERIODIC TEST III EXAM (2017-18) SUBJECT: MATHEMATICS(041) BLUE PRINT : CLASS IX Unit Chapter VSA (1 mark) SA I (2 marks) SA II (3 marks)

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

Test Corrections for Unit 1 Test

MUST READ DIRECTIONS: Read the directions located on www.koltymath.weebly.com to understand how to properly do test corrections. Ask for clarification from your teacher if there are parts that you are

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

CHAPTER TWO. 2.1 Vectors as ordered pairs and triples. The most common set of basic vectors in 3-space is i,j,k. where

40 CHAPTER TWO.1 Vectors as ordered pairs and triples. The most common set of basic vectors in 3-space is i,j,k where i represents a vector of magnitude 1 in the x direction j represents a vector of magnitude

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

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

Chapter 10. Properties of Circles

Chapter 10 Properties of Circles 10.1 Use Properties of Tangents Objective: Use properties of a tangent to a circle. Essential Question: how can you verify that a segment is tangent to a circle? Terminology:

Exercises for Unit V (Introduction to non Euclidean geometry)

Exercises for Unit V (Introduction to non Euclidean geometry) V.1 : Facts from spherical geometry Ryan : pp. 84 123 [ Note : Hints for the first two exercises are given in math133f07update08.pdf. ] 1.

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

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

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

DESIGN OF THE QUESTION PAPER

SET-I DESIGN OF THE QUESTION PAPER MATHEMATICS CLASS IX Time : 3 Hours Maximum Marks : 80 The weightage or the distribution of marks over different dimensions of the question paper shall be as follows:

Edexcel New GCE A Level Maths workbook Circle.

Edexcel New GCE A Level Maths workbook Circle. Edited by: K V Kumaran kumarmaths.weebly.com 1 Finding the Midpoint of a Line To work out the midpoint of line we need to find the halfway point Midpoint

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

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?

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

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.

1.1 Exercises, Sample Solutions

DM, Chapter, Sample Solutions. Exercises, Sample Solutions 5. Equal vectors have the same magnitude and direction. a) Opposite sides of a parallelogram are parallel and equal in length. AD BC, DC AB b)

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.

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:

the coordinates of C (3) Find the size of the angle ACB. Give your answer in degrees to 2 decimal places. (4)

. The line l has equation, 2 4 3 2 + = λ r where λ is a scalar parameter. The line l 2 has equation, 2 0 5 3 9 0 + = µ r where μ is a scalar parameter. Given that l and l 2 meet at the point C, find the

0609ge. 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

Practice Test Geometry 1. Which of the following points is the greatest distance from the y-axis? A. (1,10) B. (2,7) C. (3,5) D. (4,3) E.

April 9, 01 Standards: MM1Ga, MM1G1b Practice Test Geometry 1. Which of the following points is the greatest distance from the y-axis? (1,10) B. (,7) C. (,) (,) (,1). Points P, Q, R, and S lie on a line

(b) the equation of the perpendicular bisector of AB. [3]

HORIZON EDUCATION SINGAPORE Additional Mathematics Practice Questions: Coordinate Geometr 1 Set 1 1 In the figure, ABCD is a rhombus with coordinates A(2, 9) and C(8, 1). The diagonals AC and BD cut at

1966 IMO Shortlist. IMO Shortlist 1966

IMO Shortlist 1966 1 Given n > 3 points in the plane such that no three of the points are collinear. Does there exist a circle passing through (at least) 3 of the given points and not containing any other

PRACTICE ASSESSMENT TASK 3 655 Practice Assessment Task SET 3 Solve m - 5m + 6 \$ 0 0 Find the locus of point P that moves so that it is equidistant from the points A^-3, h and B ^57, h 3 Write x = 4t,

Euclidian Geometry Grade 10 to 12 (CAPS)

Euclidian Geometry Grade 10 to 12 (CAPS) Compiled by Marlene Malan marlene.mcubed@gmail.com Prepared by Marlene Malan CAPS DOCUMENT (Paper 2) Grade 10 Grade 11 Grade 12 (a) Revise basic results established

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

Graphs MEP Pupil Text -9, Additional Material.B Gradients of Perpendicular Lines In this section we explore the relationship between the gradients of perpendicular lines and line segments. Worked Example

CIRCLES MODULE - 3 OBJECTIVES EXPECTED BACKGROUND KNOWLEDGE. Circles. Geometry. Notes

Circles MODULE - 3 15 CIRCLES You are already familiar with geometrical figures such as a line segment, an angle, a triangle, a quadrilateral and a circle. Common examples of a circle are a wheel, a bangle,

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

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

STRAIGHT LINES EXERCISE - 3

STRAIGHT LINES EXERCISE - 3 Q. D C (3,4) E A(, ) Mid point of A, C is B 3 E, Point D rotation of point C(3, 4) by angle 90 o about E. 3 o 3 3 i4 cis90 i 5i 3 i i 5 i 5 D, point E mid point of B & D. So

PAST QUESTIONS ON VECTORS P1

PAST QUESTIONS ON VECTORS P1 1. Diagram shows a solid cylinder standing on a horizontal circular base, centre O and radius 4 units. The line BA is a diameter and the radius OC is at 90 o to OA. Points

KENDRIYA VIDYALAYA SANGATHAN, HYDERABAD REGION SAMPLE PAPER 01 F SESSING ENDING EXAM (2017-18) SUBJECT: MATHEMATICS(041) BLUE PRINT : CLASS IX Unit Chapter VSA (1 mark) SA I (2 marks) SA II (3 marks) LA

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

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

( ) = ( ) ( ) = ( ) = + = = = ( ) Therefore: , where t. Note: If we start with the condition BM = tab, we will have BM = ( x + 2, y + 3, z 5)

Chapter Exercise a) AB OB OA ( xb xa, yb ya, zb za),,, 0, b) AB OB OA ( xb xa, yb ya, zb za) ( ), ( ),, 0, c) AB OB OA x x, y y, z z (, ( ), ) (,, ) ( ) B A B A B A ( ) d) AB OB OA ( xb xa, yb ya, zb za)

Sharygin Geometry Olympiad 2013 First Round 1 Let ABC be an isosceles triangle with AB = BC. Point E lies on the side AB, and ED is the perpendicular from E to BC. It is known that AE = DE. Find DAC. 2

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

MATHEMATICS. IMPORTANT FORMULAE AND CONCEPTS for. Summative Assessment -II. Revision CLASS X Prepared by

MATHEMATICS IMPORTANT FORMULAE AND CONCEPTS for Summative Assessment -II Revision CLASS X 06 7 Prepared by M. S. KUMARSWAMY, TGT(MATHS) M. Sc. Gold Medallist (Elect.), B. Ed. Kendriya Vidyalaya GaCHiBOWli

MHR Principles of Mathematics 10 Solutions 1

Course Review Note: Length and angle measures may vary slightly due to rounding. Course Review Question Page 8 a) Let l represent the length and w represent the width, then l + w 0. n+ q b) If n represents

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

Calgary 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.

CONCURRENT LINES- PROPERTIES RELATED TO A TRIANGLE THEOREM The medians of a triangle are concurrent. Proof: Let A(x 1, y 1 ), B(x, y ), C(x 3, y 3 ) be the vertices of the triangle A(x 1, y 1 ) F E B(x,

Mathematics Revision Guides Vectors Page 1 of 19 Author: Mark Kudlowski M.K. HOME TUITION. Mathematics Revision Guides Level: GCSE Higher Tier VECTORS

Mathematics Revision Guides Vectors Page of 9 M.K. HOME TUITION Mathematics Revision Guides Level: GCSE Higher Tier VECTORS Version:.4 Date: 05-0-05 Mathematics Revision Guides Vectors Page of 9 VECTORS

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

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

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

1 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

Berkeley Math Circle, May

Berkeley Math Circle, May 1-7 2000 COMPLEX NUMBERS IN GEOMETRY ZVEZDELINA STANKOVA FRENKEL, MILLS COLLEGE 1. Let O be a point in the plane of ABC. Points A 1, B 1, C 1 are the images of A, B, C under symmetry