Class IX : Math Chapter 11: Geometric Constructions Top Concepts 1. To construct an angle equal to a given angle. Given : Any POQ and a point A.

Save this PDF as:

Size: px
Start display at page:

Download "Class IX : Math Chapter 11: Geometric Constructions Top Concepts 1. To construct an angle equal to a given angle. Given : Any POQ and a point A."

Transcription

1 1 Class IX : Math Chapter 11: Geometric Constructions Top Concepts 1. To construct an angle equal to a given angle. Given : Any POQ and a point A. Required : To construct an angle at A equal to POQ. 1. With O as centre and any (suitable) radius, draw an arc to meet OP at R and OQ at S. 2. Through A draw a line AB. 3. Taking A as centre and same radius (as in step 1), draw an arc to meet AB at D. 4. Measure the segment RS with compasses. 5. With d as centre and radius equal to RS, draw an arc to meet the previous arc at E. 6. Join AE and produce it to C, then BAC is the required angle equal to POQ 2. To bisect a given angle. Given : Any POQ Required : To bisect POQ. 1. With O as centre and any (suitable) radius, draw an arc to meet OP at R and OQ at S.

2 2 2. With R as centre and any suitable radius (not necessarily) equal to radius of step 1 (but > 1 2 RS), draw an arc. Also, with S as centre and same radius draw another arc to meet the previous arc at T. 3. Join OT and produce it, then OT is the required bisector of POQ. 3. To construct angles of 60, 30, 120, 90, 45 (i) To construct an angle of Draw any line OP. 2. With O as centre and any suitable radius, draw an arc to meet OP at R. 3. With R as centre and same radius (as in step 2), draw an arc to meet the previous arc at S. 4. Join OS and produce it to Q, then POQ = 60.

3 3 (ii) To construct an angle of 30 Steps of Construction 1. Construct POQ = 60 (as above). 2. Bisect POQ (as in construction 2). Let OT be the bisector of POQ, then POT = 30 (iii) To construct an angle of Draw any line OP. 2. With O as centre and any suitable radius, draw an arc to meet OP at R. 3. With R as centre and same radius (as in step 2), draw an arc to meet the previous arc at T. With T as centre and same radius, draw another arc to cut the first arc at S. 4. Join OS and produce it to Q, then POQ = 120.

4 4 (iv) To construct an angle of 90 Steps of Construction 1. Construct POQ = 60 (as in construction 3(i)). 2. Construct POV = 120 (as above). 3. Bisect QOV (as in construction 2). Let OU be the bisector of QOV, then POU = 90. (v) To construct an angle of 45 Steps of Construction 1. Construct AOP = 90 (as above). 2. Bisect AOP (as in construction 2). Let OQ be the bisector of AOP, then AOQ = 45

5 5 4. To bisect a given line segment. Given : Any line segment AB. Required : To bisect line segment AB. 1. At A, construct any suitable angle BAC. 2. At B, construct ABD = BAC on the other side of the line AB. 3. With A as centre and any suitable radius, draw an arc to meet AC at E. 4. From BD, cut off BF = AE. 5. Join EF to meet AB at G, then EG is a bisector of the line segment AB and G is mid point of AB. (ii) To divided a given line segment in a number of equal part.

6 6 5. Divided a line segment AB of length 8 cm into 4 equal part. Given : A line segment AB of length 8 cm. Required : To divide line segment 8 cm into 4 equal parts. 1. Draw lien segment AB = 8 cm. 2. At A, construct any suitable angle BAX. 3. At B, construct ABY = BAX on the other side of the line AB. 4. From AX, cut off 4 equal distances at the points C, D, E and F such that AC = CD = DE = EF. 5. With the same radius, cut off 4 equal distances along BY at the points H, I, J and K such that BH = HI = IJ = JK. 6. Join AK, CJ, DI, EH and FB. Let CJ, DI and EH meet the line segment AB at the points M, N and O respectively. Then, M, N and O are the points of division of AB such that AM = MN = NO = OB. 6. To draw a perpendicular bisector of a line segment. Given : Any line segment PQ. Required : To draw a perpendicular bisector of lien segment PQ.

7 7 1. With P as centre and any line suitable radius draw arcs, one on each side of PQ. 2. With Q as centre and same radius (as in step 1), draw two more arcs, one on each side of PQ cutting the previous arcs at A and B. 3. Join AB to meet PQ at M, then AB bisects PQ at M, and is perpendicular to PQ, Thus, AB is the required perpendicular bisector of PQ. 7. To construct an equilateral triangle when one of its side is given. E.g.: Construct and equilateral triangle whose each side is 5 cm. Given : Each side of an equilateral triangle is 5 cm. Required : To construct the equilateral triangle. 1. Draw any line segment AB = 5 cm. 2. With A as centre and radius 5 cm draw an arc. 3. With B as centre and radius 5 cm draw an arc to cut the previous arc at C. 4. Join AC and BC. Then ABC is the required triangle.

8 8 8. To construct an equilateral triangle when its altitude is given. E.g.: Construct an equilateral triangle whose altitude is 4 cm. 1. Draw any line segment PQ. 2. Take an point D on PQ and At D, construct perpendicular DR to PQ. From DR, cut off DA = 4 cm. 3. At A, construct DAS = DAT = 1 60 = 30 on either side of 2 AD. Let AS and AT meet PQ at points B and C respectively. Then, ABC is the required equilateral triangle. 9. Construction of a triangle, given its Base, Sum of the other Two sides and one Base Angle.

9 9 E.g Construct a triangle with base of length 5 cm, the sum of the other two sides 7 cm and one base angle of 60. Given: In ABC, base BC = 5 cm, AB + AC = 7 cm and ABC = 60 Required : To construct the ABC. 1. Draw BC = 5 cm. 2. At B, construct CBX = From BX, cut off BD = 7 cm. 4. Join CD. 5. Draw the perpendicular bisector of CD, intersecting BD at a point A. 6. Join AC. Then, ABC is the required triangle. 10. Construction of a triangle, Given its Base, Difference of the Other Two Sides and one Base Angle. Eg: Construct a triangle with base of length 7.5 cm, the difference of the other two sides 2.5 cm, and one base angle of 45 Given : In ABC, base BC = 7.5 cm, the difference of the other two sides, AB AC or AC AB = 2.5 cm and one base angle is 45. Required : To construct the ABC, CASE (i) AB AC = 2.5 cm.

10 10 1. Draw BC = 7.5 cm. 2. At B, construct CBX = From BX, cut off BD = 2.5 cm. 4. Join CD. 5. Draw the perpendicular bisector RS of CD intersecting BX at a point A. 6. Join AC. Then, ABC is the required triangle. CASE (ii) AC AB = 2.5 cm 1. Draw BC = 7.5 cm. 2. At B, construct CBX = 45 and produce XB to form a line XBX. 3. From BX, cut off BD = 2.5 cm. 4. Join CD. 5. Draw perpendicular bisector RS of CD intersecting BX at a point A. 6. Join AC. Then, ABC is the required triangle.

11 11 e 11. Construction of a Triangle of Given Perimeter and Base Angles. Construct a triangle with perimeter 11.8 cm and base angles 60 and 45. Given : In ABC, AB+BC+CA = 11.8 cm, B = 60 & C = 45. Required : To construct the ABC. 1. Draw DE = 11.8 cm. 2. At D, construct EDP = 1 2 of 60 = 30 and at E, construct DEQ = 1 2 of 45 = Let DP and EQ meet at A. 4. Draw perpendicular bisector of AD to meet DE at B. 5. Draw perpendicular bisector of AE to meet DE at C. 6. Join AB and AC. Then, ABC is the required triangle.

12 12

(A) 50 (B) 40 (C) 90 (D) 75. Circles. Circles <1M> 1.It is possible to draw a circle which passes through three collinear points (T/F)

Circles 1.It is possible to draw a circle which passes through three collinear points (T/F) 2.The perpendicular bisector of two chords intersect at centre of circle (T/F) 3.If two arcs of a circle

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

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

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

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.

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!!! " "

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)

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

It is known that the length of the tangents drawn from an external point to a circle is equal.

CBSE -MATHS-SET 1-2014 Q1. The first three terms of an AP are 3y-1, 3y+5 and 5y+1, respectively. We need to find the value of y. We know that if a, b and c are in AP, then: b a = c b 2b = a + c 2 (3y+5)

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,

SUMMATIVE ASSESSMENT I, IX / Class IX

I, 0 SUMMATIVE ASSESSMENT I, 0 0 MATHEMATICS / MATHEMATICS MATHEMATICS CLASS CLASS - IX - IX IX / Class IX MA-0 90 Time allowed : hours Maximum Marks : 90 (i) (ii) 8 6 0 0 (iii) 8 (iv) (v) General Instructions:

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,

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

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?

2012 Mu Alpha Theta National Convention Theta Geometry Solutions ANSWERS (1) DCDCB (6) CDDAE (11) BDABC (16) DCBBA (21) AADBD (26) BCDCD SOLUTIONS

01 Mu Alpha Theta National Convention Theta Geometry Solutions ANSWERS (1) DCDCB (6) CDDAE (11) BDABC (16) DCBBA (1) AADBD (6) BCDCD SOLUTIONS 1. Noting that x = ( x + )( x ), we have circle is π( x +

Unit 1. GSE Analytic Geometry EOC Review Name: Units 1 3. Date: Pd:

GSE Analytic Geometry EOC Review Name: Units 1 Date: Pd: Unit 1 1 1. Figure A B C D F is a dilation of figure ABCDF by a scale factor of. The dilation is centered at ( 4, 1). 2 Which statement is true?

SUMMATIVE ASSESSMENT - I (2012) MATHEMATICS CLASS IX. Time allowed : 3 hours Maximum Marks :90

SUMMATIVE ASSESSMENT - I (2012) MATHEMATICS CLASS IX Time allowed : 3 hours Maximum Marks :90 General Instructions: i. All questions are compulsory. ii. The question paper consists of 34 questions divided

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

TOPIC 4 Line and Angle Relationships. Good Luck To. DIRECTIONS: Answer each question and show all work in the space provided.

Good Luck To Period Date DIRECTIONS: Answer each question and show all work in the space provided. 1. Name a pair of corresponding angles. 1 3 2 4 5 6 7 8 A. 1 and 4 C. 2 and 7 B. 1 and 5 D. 2 and 4 2.

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

1.4 Midpoints and Bisectors

www.ck12.org Chapter 1. Basics of Geometry 1.4 Midpoints and Bisectors Learning Objectives Identify the midpoint of line segments. Identify the bisector of a line segment. Understand and the Angle Bisector

Q1. The sum of the lengths of any two sides of a triangle is always (greater/lesser) than the length of the third side. askiitians

Class: VII Subject: Math s Topic: Properties of triangle No. of Questions: 20 Q1. The sum of the lengths of any two sides of a triangle is always (greater/lesser) than the length of the third side. Greater

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

1 Solution of Final. Dr. Franz Rothe December 25, Figure 1: Dissection proof of the Pythagorean theorem in a special case

Math 3181 Dr. Franz Rothe December 25, 2012 Name: 1 Solution of Final Figure 1: Dissection proof of the Pythagorean theorem in a special case 10 Problem 1. Given is a right triangle ABC with angle α =

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

(Chapter 10) (Practical Geometry) (Class VII) Question 1: Exercise 10.1 Draw a line, say AB, take a point C outside it. Through C, draw a line parallel to AB using ruler and compasses only. Answer 1: To

Question Bank Tangent Properties of a Circle

Question Bank Tangent Properties of a Circle 1. In quadrilateral ABCD, D = 90, BC = 38 cm and DC = 5 cm. A circle is inscribed in this quadrilateral which touches AB at point Q such that QB = 7 cm. Find

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

1. 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

0112ge. 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

UNIT 1: SIMILARITY, CONGRUENCE, AND PROOFS. 1) Figure A'B'C'D'F' is a dilation of figure ABCDF by a scale factor of 1. 2 centered at ( 4, 1).

1) Figure A'B'C'D'F' is a dilation of figure ABCDF by a scale factor of 1. 2 centered at ( 4, 1). The dilation is Which statement is true? A. B. C. D. AB B' C' A' B' BC AB BC A' B' B' C' AB BC A' B' D'

AREAS 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

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

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

(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

CBSE 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

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

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,

Class : IX(CBSE) Worksheet - 1 Sub : Mathematics Topic : Number system

Class : IX(CBSE) Worksheet - Sub : Mathematics Topic : Number system I. Solve the following:. Insert rational numbers between. Epress 57 65 in the decimal form. 8 and.. Epress. as a fraction in the simplest

Geometry 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)

SMT 2018 Geometry Test Solutions February 17, 2018

SMT 018 Geometry Test Solutions February 17, 018 1. Consider a semi-circle with diameter AB. Let points C and D be on diameter AB such that CD forms the base of a square inscribed in the semicircle. Given

Properties of Isosceles and Equilateral Triangles

Properties of Isosceles and Equilateral Triangles In an isosceles triangle, the sides and the angles of the triangle are classified by their position in relation to the triangle s congruent sides. Leg

Geometry Arcs and Chords. Geometry Mr. Austin

10.2 Arcs and Chords Mr. Austin Objectives/Assignment Use properties of arcs of circles, as applied. Use properties of chords of circles. Assignment: pp. 607-608 #3-47 Reminder Quiz after 10.3 and 10.5

Suppose E is the centre of the line joining the feet of the two towers i.e. BD.

CLASS X : MATH SOLUTIONS Q1. It is given that x = - 1 is the solution of the quadratic equation 3x +kx-3 = 0. 3 1 1 + k (- ) - 3 =0 3 4 - k - 3=0 k = 3 3 = 9 4 4 Hence, the value of k is 9 4 Q. Let AB

1. Matrices and Determinants

Important Questions 1. Matrices and Determinants Ex.1.1 (2) x 3x y Find the values of x, y, z if 2x + z 3y w = 0 7 3 2a Ex 1.1 (3) 2x 3x y If 2x + z 3y w = 3 2 find x, y, z, w 4 7 Ex 1.1 (13) 3 7 3 2 Find

0116ge. Geometry Regents Exam RT and SU intersect at O.

Geometry Regents Exam 06 06ge What is the equation of a circle with its center at (5, ) and a radius of 3? ) (x 5) + (y + ) = 3 ) (x 5) + (y + ) = 9 3) (x + 5) + (y ) = 3 4) (x + 5) + (y ) = 9 In the diagram

P-RMO 017 NATIONAL BOARD FOR HIGHER MATHEMATICS AND HOMI BHABHA CENTRE FOR SCIENCE EDUCATION TATA INSTITUTE OF FUNDAMENTAL RESEARCH Pre-REGIONAL MATHEMATICAL OLYMPIAD, 017 TEST PAPER WITH SOLUTION & ANSWER

( ) = ( ) ( ) = ( ) = + = = = ( ) 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)

So, eqn. to the bisector containing (-1, 4) is = x + 27y = 0

Q.No. The bisector of the acute angle between the lines x - 4y + 7 = 0 and x + 5y - = 0, is: Option x + y - 9 = 0 Option x + 77y - 0 = 0 Option x - y + 9 = 0 Correct Answer L : x - 4y + 7 = 0 L :-x- 5y

Prepared by: M. S. KumarSwamy, TGT(Maths) Page

Prepared by: M S KumarSwamy, TGT(Maths) Page - 119 - CHAPTER 10: VECTOR ALGEBRA QUICK REVISION (Important Concepts & Formulae) MARKS WEIGHTAGE 06 marks Vector The line l to the line segment AB, then a

1-2 Measuring and Constructing Segments

1-2 Measuring and Constructing Segments Warm Up Lesson Presentation Lesson Quiz Objectives Use length and midpoint of a segment. Construct midpoints and congruent segments. Vocabulary coordinate midpoint

Basic Trigonometry. Trigonometry deals with the relations between the sides and angles of triangles.

Basic Trigonometry Trigonometry deals with the relations between the sides and angles of triangles. A triangle has three sides and three angles. Depending on the size of the angles, triangles can be: -

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

2007 Shortlist JBMO - Problems

Chapter 1 007 Shortlist JBMO - Problems 1.1 Algebra A1 Let a be a real positive number such that a 3 = 6(a + 1). Prove that the equation x + ax + a 6 = 0 has no solution in the set of the real number.

SUMMATIVE ASSESSMENT II SAMPLE PAPER I MATHEMATICS

SUMMATIVE ASSESSMENT II SAMPLE PAPER I MATHEMATICS Class: IX Time: 3-3 ½ hours M.Marks:80 General Instructions: 1. All questions are compulsory 2. The question paper consists of 34 questions divided into

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

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 =,

Chapter - 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

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)

International Mathematics TOURNAMENT OF THE TOWNS

International Mathematics TOURNAMENT OF THE TOWNS Senior A-Level Paper Fall 2011 1 Pete has marked at least 3 points in the plane such that all distances between them are different A pair of marked points

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

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

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.

The Canadian Mathematical Society in collaboration with The CENTRE for EDUCATION in MATHEMATICS and COMPUTING presents the Canadian Open Mathematics Challenge Wednesday, November, 006 Supported by: Solutions

Midpoints and Bisectors

Midpoints and Bisectors Say Thanks to the Authors Click http://www.ck12.org/saythanks (No sign in required) To access a customizable version of this book, as well as other interactive content, visit www.ck12.org

Geometry Problem Solving Drill 08: Congruent Triangles

Geometry Problem Solving Drill 08: Congruent Triangles Question No. 1 of 10 Question 1. The following triangles are congruent. What is the value of x? Question #01 (A) 13.33 (B) 10 (C) 31 (D) 18 You set

POLYNOMIALS ML 5 ZEROS OR ROOTS OF A POLYNOMIAL. A real number α is a root or zero of polynomial f(x) = x + a x + a x +...

POLYNOMIALS ML 5 ZEROS OR ROOTS OF A POLYNOMIAL n n 1 n an n 1 n 1 + 0 A real number α is a root or zero of polynomial f(x) = x + a x + a x +... + a x a, n n an n 1 n 1 0 = if f (α) = 0. i.e. α + a + a

Maharashtra State Board Class X Mathematics Geometry Board Paper 2015 Solution. Time: 2 hours Total Marks: 40

Maharashtra State Board Class X Mathematics Geometry Board Paper 05 Solution Time: hours Total Marks: 40 Note:- () Solve all questions. Draw diagrams wherever necessary. ()Use of calculator is not allowed.

VECTORS 1. ADDITION OF VECTORS. SCALAR PRODUCT OF VECTORS 3. VECTOR PRODUCT OF VECTORS 4. SCALAR TRIPLE PRODUCT 5. VECTOR TRIPLE PRODUCT 6. PRODUCT OF FOUR VECTORS ADDITIONS OF VECTORS Definitions and

LINES and ANGLES CLASS IX

LINES and ANGLES CLASS IX Questions From CBSE Examination Papers 1. In the given figure, the value of x which makes POQ a straight line is: 6. The complementary angles are in the ratio 1 : 5. Find the

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

Foundations of Neutral Geometry

C H A P T E R 12 Foundations of Neutral Geometry The play is independent of the pages on which it is printed, and pure geometries are independent of lecture rooms, or of any other detail of the physical

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:

QUESTION BANK ON. CONIC SECTION (Parabola, Ellipse & Hyperbola)

QUESTION BANK ON CONIC SECTION (Parabola, Ellipse & Hyperbola) Question bank on Parabola, Ellipse & Hyperbola Select the correct alternative : (Only one is correct) Q. Two mutually perpendicular tangents

Chapter 1 Coordinates, points and lines

Cambridge Universit Press 978--36-6000-7 Cambridge International AS and A Level Mathematics: Pure Mathematics Coursebook Hugh Neill, Douglas Quadling, Julian Gilbe Ecerpt Chapter Coordinates, points and

Geometry Arcs and Chords. Geometry Mr. Peebles Spring 2013

10.2 Arcs and Chords Geometry Mr. Peebles Spring 2013 Bell Ringer: Solve For r. B 16 ft. A r r 8 ft. C Bell Ringer B 16 ft. Answer A r r 8 ft. C c 2 = a 2 + b 2 Pythagorean Thm. (r + 8) 2 = r 2 + 16 2

Hanoi Open Mathematical Competition 2017

Hanoi Open Mathematical Competition 2017 Junior Section Saturday, 4 March 2017 08h30-11h30 Important: Answer to all 15 questions. Write your answers on the answer sheets provided. For the multiple choice

C5WEBZL SUMMATIVE ASSESSMENT II MATHEMATICS Class IX Time allowed : 3 hours Maximum Marks : 90

C5WEBZL SUMMATIVE ASSESSMENT II MATHEMATICS Class IX Time allowed : hours Maximum Marks : 90 General Instructions : (i) All questions are compulsory. (ii) The question paper consists of 1 questions divided

UNIT 1: SIMILARITY, CONGRUENCE, AND PROOFS. 1) Figure A'B'C'D'F' is a dilation of figure ABCDF by a scale factor of 1. 2 centered at ( 4, 1).

EOCT Practice Items 1) Figure A'B'C'D'F' is a dilation of figure ABCDF by a scale factor of 1. 2 centered at ( 4, 1). The dilation is Which statement is true? A. B. C. D. AB B' C' A' B' BC AB BC A' B'

EXERCISE 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.

5-1 Perpendicular and Angle Bisectors

5-1 Perpendicular and Angle Bisectors Warm Up Lesson Presentation Lesson Quiz Geometry Warm Up Construct each of the following. 1. A perpendicular bisector. 2. An angle bisector. 3. Find the midpoint and

5. Introduction to Euclid s Geometry

5. Introduction to Euclid s Geometry Multiple Choice Questions CBSE TREND SETTER PAPER _ 0 EXERCISE 5.. If the point P lies in between M and N and C is mid-point of MP, then : (A) MC + PN = MN (B) MP +

2007 Hypatia Contest

Canadian Mathematics Competition An activity of the Centre for Education in Mathematics and Computing, University of Waterloo, Waterloo, Ontario 007 Hypatia Contest Wednesday, April 18, 007 Solutions c

First day. 8 grade 8.1. Let ABCD be a cyclic quadrilateral with AB = = BC and AD = CD. ApointM lies on the minor arc CD of its circumcircle. The lines BM and CD meet at point P, thelinesam and BD meet

KENDRIYA VIDYALAYA SANGATHAN, HYDERABAD REGION SAMPLE PAPER 01 FOR HALF YEARLY EXAM (017-18) SUBJECT: MATHEMATICS(041) BLUE PRINT FOR HALF YEARLY EXAM: CLASS IX Chapter VSA (1 mark) SA I ( marks) SA II

Math Wrangle Practice Problems

Math Wrangle Practice Problems American Mathematics Competitions November 19, 2010 1. Find the sum of all positive two-digit integers that are divisible by each of their digits. 2. A finite set S of distinct

SURA's Guides for 3rd to 12th Std for all Subjects in TM & EM Available. MARCH Public Exam Question Paper with Answers MATHEMATICS

SURA's Guides for rd to 1th Std for all Subjects in TM & EM Available 10 th STD. MARCH - 017 Public Exam Question Paper with Answers MATHEMATICS [Time Allowed : ½ Hrs.] [Maximum Marks : 100] SECTION -

Class 9 Geometry-Overall

ID : in-9-geometry-overall [1] Class 9 Geometry-Overall For more such worksheets visit www.edugain.com Answer t he quest ions (1) AB is diameter of the circle. If A and B are connected to E, circle is

S Group Events G1 a 47 G2 a *2

Answers: (003-0 HKMO Final Events) Created by: Mr. Francis Hung Last updated: 9 September 07 Individual Events I a 6 I P I3 a I a IS P 8 b + Q 6 b 9 b Q 8 c 7 R 6 c d S 3 d c 6 R 7 8 d *33 3 see the remark

Circle geometry investigation: Student worksheet

Circle geometry investigation: Student worksheet http://topdrawer.aamt.edu.au/geometric-reasoning/good-teaching/exploringcircles/explore-predict-confirm/circle-geometry-investigations About these activities

Answers: ( HKMO Final Events) Created by Mr. Francis Hung Last updated: 13 December Group Events

Individual Events SI a 900 I1 P 100 I k 4 I3 h 3 I4 a 18 I5 a 495 b 7 Q 8 m 58 k 6 r 3 b p R 50 a m 4 M 9 4 x 99 q 9 S 3 b 3 p Group Events 15 16 w 1 Y 109 SG p 75 G6 x 15 G7 M 5 G8 S 7 G9 p 60 G10 n 18

CONGRUENCE 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

BC Exam Solutions Texas A&M High School Math Contest October 24, p(1) = b + 2 = 3 = b = 5.

C Exam Solutions Texas &M High School Math Contest October 4, 01 ll answers must be simplified, and If units are involved, be sure to include them. 1. p(x) = x + ax + bx + c has three roots, λ i, with

Honors 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

USA Mathematical Talent Search Solutions to Problem 5/3/16

Solutions to Problem 5//16 5//16. Consider an isosceles triangle ABC with side lengths AB = AC = 10 2 and BC = 10. Construct semicircles P, Q, and R with diameters AB, AC, BC respectively, such that the

Name. 9. Find the diameter and radius of A, B, and C. State the best term for the given figure in the diagram.

Name LESSON 10.1 State the best term for the given figure in the diagram. 9. Find the diameter and radius of A, B, and C. 10. Describe the point of intersection of all three circles. 11. Describe all the

2016 State Mathematics Contest Geometry Test

2016 State Mathematics Contest Geometry Test In each of the following, choose the BEST answer and record your choice on the answer sheet provided. To ensure correct scoring, be sure to make all erasures

OUR OWN HIGH SCHOOL, AL WARQA A, DUBAI CMLSTQT SUMMATIVE ASSESSMENT II ( ) MATHEMATICS Class IX Time allowed : 3 hours Maximum Marks : 90

OUR OWN HIGH SCHOOL, AL WARQA A, DUBAI CMLSTQT SUMMATIVE ASSESSMENT II (05-06) MATHEMATICS Class IX Time allowed : hours Maximum Marks : 90 General Instructions : (i) All questions are compulsory. (ii)