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

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

Triangles. 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.

Class IX Chapter 8 Quadrilaterals Maths

Class IX Chapter 8 Quadrilaterals Maths

Chapter 8 Similar Triangles

Similarity of Triangle

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

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

9 th CBSE Mega Test - II

Triangles. Chapter Flowchart. The Chapter Flowcharts give you the gist of the chapter flow in a single glance.

Downloaded from

Class IX Chapter 7 Triangles Maths. Exercise 7.1 Question 1: In quadrilateral ACBD, AC = AD and AB bisects A (See the given figure).

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?

Triangle Congruence and Similarity Review. Show all work for full credit. 5. In the drawing, what is the measure of angle y?

Class IX Chapter 7 Triangles Maths

CHAPTER 7 TRIANGLES. 7.1 Introduction. 7.2 Congruence of Triangles

SHW 1-01 Total: 30 marks

CONGRUENCE OF TRIANGLES

RD Sharma Solutions for Class 10 th

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

BOARD QUESTION PAPER : MARCH 2016 GEOMETRY

Geometry Midterm Exam Review 3. Square BERT is transformed to create the image B E R T, as shown.

Visit: ImperialStudy.com For More Study Materials Class IX Chapter 12 Heron s Formula Maths

Page 1 of 15. Website: Mobile:

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

ISSUED BY KENDRIYA VIDYALAYA - DOWNLOADED FROM TRIANGLES KEY POINTS

Created by T. Madas 2D VECTORS. Created by T. Madas

TRIANGLE EXERCISE 6.4

Mathematics. Exercise 6.4. (Chapter 6) (Triangles) (Class X) Question 1: Let and their areas be, respectively, 64 cm 2 and 121 cm 2.


Class 7 Lines and Angles

2012 GCSE Maths Tutor All Rights Reserved

VAISHALI EDUCATION POINT (QUALITY EDUCATION PROVIDER)

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

EXERCISE 10.1 EXERCISE 10.2


Chapter 7. Geometric Inequalities

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

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.

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

Properties of the Circle

3. AD is a diameter of a circle and AB is a chord. If AD = 34 cm, AB = 30 cm, the distance of AB from the centre of the circle is:

Class-IX CBSE Latest Pattern Sample Paper {Mathematics}

Class IX - NCERT Maths Exercise (10.1)

IB MYP Unit 6 Review

Revision Question Bank

0811ge. Geometry Regents Exam

Properties of Isosceles and Equilateral Triangles

LLT Education Services

1 / 23

Geometry Honors Review for Midterm Exam

AREAS OF PARALLELOGRAMS AND TRIANGLES

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

SSC CGL Tier 1 and Tier 2 Program

Chapter - 7. (Triangles) Triangle - A closed figure formed by three intersecting lines is called a triangle. A

( )( ) PR PQ = QR. Mathematics Class X TOPPER SAMPLE PAPER-1 SOLUTIONS. HCF x LCM = Product of the 2 numbers 126 x LCM = 252 x 378

Chapter 3 Cumulative Review Answers

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

CLASS IX MID TERM EXAMINATION ( ) Subject: MATHS SOLUTIONS. Set B-2. TIME :3hrs MAX.MARKS: 80

Nozha Directorate of Education Form : 2 nd Prep


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

Mathematics. A basic Course for beginers in G.C.E. (Advanced Level) Mathematics

Math 3 Review Sheet Ch. 3 November 4, 2011

CBSE MATHEMATICS (SET-2)_2019

Honors Geometry Mid-Term Exam Review

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

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

Part (1) Second : Trigonometry. Tan

Postulates and Theorems in Proofs

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

SUMMATIVE ASSESSMENT-1 SAMPLE PAPER (SET-2) MATHEMATICS CLASS IX

Question 1 ( 1.0 marks) places of decimals? Solution: Now, on dividing by 2, we obtain =

0609ge. Geometry Regents Exam AB DE, A D, and B E.

THEOREMS WE KNOW PROJECT

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

Euclidian Geometry Grade 10 to 12 (CAPS)

9. Areas of Parallelograms and Triangles

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.

Lesson. Warm Up deductive 2. D. 3. I will go to the store; Law of Detachment. Lesson Practice 31

Udaan School Of Mathematics Class X Chapter 10 Circles Maths

0112ge. Geometry Regents Exam Line n intersects lines l and m, forming the angles shown in the diagram below.

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

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.

AREAS OF PARALLELOGRAMS AND TRIANGLES

Midterm Review Packet. Geometry: Midterm Multiple Choice Practice

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.

3D GEOMETRY. 3D-Geometry. If α, β, γ are angle made by a line with positive directions of x, y and z. axes respectively show that = 2.

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

Geometry Problem Solving Drill 08: Congruent Triangles


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)

(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)

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

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

GEOMETRY Teacher: Mrs. Flynn Topic: Similarity. Teacher: Mrs. Flynn Topic: Similarity

Class Notes on Pythagoras Theorem (Chapter 12) Chapter 12 English Version

1. If two angles of a triangle measure 40 and 80, what is the measure of the other angle of the triangle?

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

Transcription:

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 FOR TRIANGLES TO BE CONGRUENT When three sides are equal When two sides and their included angles are equal When two angles and one side is equal CONDITIONS FOR TRIANGLES TO BE CONGRUENT When three sides are equal AB=QR BC =PQ AC = PR ABC RQP. Reference: SSS When two sides and their included angles are equal If AB = PQ BC = QR, and B = Q Then ABC PQR. Reference: SAS

When two angles and one side is equal If B = Q A = P and AB = PQ Then ABC PQR. Reference: ASA Similiarity: Two figures are said to be similar when they have same shape. Example : Proto type of a building and the actual building are similar When we deal with two dimension figures we use similarity to prove the given figures are similar. Similarity can be proved by conditions given below CONDITIONS FOR TRIANGLES TO BE SIMILAR Three Angles are Equal If A = X B = Y and C = Z then ABC ~ XYZ. Reference: AAA WHEN TWO TRIANGLE ARE SIMILAR Three Angles are Equal If ABC ~ XYZ. Then, BC = AC = AB YZ XZ XY Reference: By similar Triangles

BASIC THEOREM OF PROPORTIONALITY Thales Theorem: If a straight line divides any two sides of a triangle in the same ratio, then the straight line is parallel to the third side of the triangle. If, AD = AE DB EC Then DE BC MID POINT THEOREM If the midpoints of two adjacent sides of a triangle are joined by a line segment, then this segment is parallel and half of the third side, If, AD = BD & AE = CE, Then DE BC and DE = 1 2 BC THEOREM The areas of similar triangles are proportional to the squares of corresponding sides, altitude, or median. Then, ar ( ABC) = AB2 = AC2 = BC2 = AM2 or ( DEF) DE 2 DF 2 EF 2 DN 2 SOME IMPORTANT CONCLUSION APB is a right triangle with P = 90. PN is perpendicular, drawn from P, to the hypotenuse AB. Then, (i) PN 2 = AN.NB (ii) AP 2 = AN.AB (iii) BP 2 = BN.BA

Ex: In Fig. AB and DE are perpendiculars to BC. If AB = 9 cm, DE = 3 cm and AC = 24 cm. Calculate AD. Sol: In similar ABC and CDE AB DE = AC DC 9 = 24 3 DC DC = 8 cm AD = AC DC = 24 8 = 16 cm Ex: In Fig. AQ and PB are perpendiculars to AB. If AQ = 14 cm, PB = 3.5 cm and AO = 6 cm. Calculate OP.? (Approx.) Sol: In similar OAQ and OBP, OA = OB AQ PB 6 = OB 14 3.5, OB = 1.5 cm In right triangle OBP, OP 2 = OB 2 + PB 2 OP 2 = 2.25 + 12.25 = 14.5, OP = 3.8 cm Ex: The sides A, BC of a trapezium ABCD are parallel & the diagonals AC, BD meet at O. The area of the BOB = 5cm 2, and the ratio between OA : OC = 3 :5. Calculate the area of triangle AOD?

Sol: In similar AOD and COB Then, ar( AOD) ar( BOC) =AO2 BO 2 ar( AOD) 5 = 9 25 ar( AOD) = 9 = 1.8 cm2 5 Ex: In the given figure, ABC and CEF are two triangles where BA is parallel to CE, and AF: AC = 5: 8. Find AD, if CE = 6 cm Sol: In similar triangles ADF and CEF AD = AF CE CF AD = 5x 6 3x AD = 10 cm (By theorem) (AF = 5x, AC = 8x, CF= 3x) Ex: BE and CF be the two medians of a ABC and G be their intersection. Also let EF cut AG at O. Then find AO: OG? Sol: In similar AOE and ADC, OA AD = AE AC = 1 2 But AD AG = 3 2 OA AD x AD AG = 1 2 3 2 4 OA = 3 AG 4 OA = 3(OA + OG) OA = 3 OG (E is the mid point of AC) OA : OG = 3 : 1

Ex: In PQR, S and T are points on side PR and PQ respectively such that PQR = PST. If PT = 5cm, PS = 3cm and TQ = 3cm, then find the length of SR? Sol: In PQR and PST, PQR = PST and P is common PQ = PR PS PT 8 3 = PR 5 PR = 8 X 5 3 = 40 40, then SR = - 3 3 3 =31 9 Ex: In ABC it is given that AB = 6cm, AC = 8cm AD is the angle bisector of BAC. Then find the ratio of BD and DC? Sol: In ABC, AD is angle bisector of BAC so BD = AB 6 DC AC 8 BD : AC = 3 : 4 Ex: In ABC D is point on BC in such that AD BC and E is point on AD in such that AE : ED = 5:1, If BAD = 30 0 and tan ( ACB) = 6 tan ( DBE) then find the ( ACB)? Sol. tan ( ACB) = 6 tan ( DBE) ( ACB) = 6 ( DBE) DBA = 180 0 30 0 90 0 = 60 0 In ADB & EDB

DBA = AD 600 = 6 DBE ED DBE 1 DBE = 10 0 ACB = 6 x 10 = 60 0 Ex: A straight line parallel to BC of ABC intersects AB and AC at point P and Q given. AP = QC, PB =4 units and AQ = 9 units, then find the length of AP? Sol: In ABC, PQ BC AP AB = AQ AC AP+PB AP = AQ+QC AQ PB = QC AP AQ = AP AQ Then AP 2 =PB x AQ = 4 x 9 = 36 AP = 6 Ex: In the given figure BAD = CAD, AB= 4 cm., AC = 5.2 BD = 3 cm then find length of the BC? Sol: BAD = CAD AB AC = BD CD 4 5.2 = 3 CD CD= 3.9 BC = 3.9 + 3 = 6.9 cm

Ex: In ABC, D and E point onn AB and AC in such that AD = 1 3 AB and AE = 1 AC. If BC = 15cm then find the length of DE? 3 Sol: AD AC = AE AC = DE BC =1 3 DE 15 = 1 3 DE= 5cm Ex: In ABC, ab = 10 cm, BC = 8cm, CA = 6cm and M is mid point of BC is a point on AC in such that MN AB, then find the area of trapezium ABMN? Sol: In ABC MN AB So, CN = NM CA AB 3 = NM 6 10 NM = 5 cm area of ABC = 24 cm 2 area of CNM = 6 So area trapezium ABMN = 24 6 = 18cm 2

2024 © sciencedocbox.com Privacy Policy | Terms of Service | Feedback