Last updated at May 29, 2018 by Teachoo

Transcript

Ex 6.4, 5 D, E and F are respectively the mid-points of sides AB, BC and CA of ΞABC. Find the ratio of the areas of ΞDEF and ΞABC. Given: Ξ ABC & D,E,F mid-points of AB,BC & CA respectively To find: (ππ βπ·πΈπΉ)/(ππ βπ΄π΅πΆ) Note: Since we need to find ratio of area of ΞDEF and ΞABC. We first need to prove these triangles are similar Solution: We know that line joining mid-points of two sides of a triangle is parallel to the 3rd side In ΞABC , D and F are mid-points of AB and AC resp., β΄ DF β₯ BC So, DF β₯ BE also Similarly, E and F are mid-points of BC and AC resp. EF β₯ AB Hence, EF β₯ DB From (1) & (2) DF β₯ BE & FE β₯ DB Therefore, opposite sides of quadrilateral is parallel DBEF is a parallelogram DBEF is a parallelogram Now we know that , in parallelogram, opposite angle are equal Hence β DFE =β ABC Similarity, we can prove DECF is a parallelogram In a parallelogram, opposite angles are equal Hence, β EDF= β ACB Now , in ΞEDF and ΞABC β DFE =β ABC β EDF= β ACB By using AA similarity criterion Ξ DEF βΌ Ξ ABC We know that if two triangles are similar, the ratio of their area is always equal to the square of the ratio of their corresponding side β΄ (ππ βπ·πΈπΉ)/(ππ βπ΄π΅πΆ) = π·πΈ2/π΄πΆ2 (ππ βπ·πΈπΉ)/(ππ βπ΄π΅πΆ) = πΉπΆ2/π΄πΆ2 (π΄πππ ππ βπ·πΈπΉ)/(π΄πππ ππ βπ΄π΅πΆ)=( π΄πΆ/2 )^2/(π΄πΆ)2 (π΄πππ ππ βπ·πΈπΉ)/(π΄πππ ππ βπ΄π΅πΆ)=((π΄πΆ)2/4)/(π΄πΆ)2 (π΄πππ ππ βπ·πΈπΉ)/(π΄πππ ππ βπ΄π΅πΆ)=(1/4)/1 Hence , (π΄πππ ππ β π·πΈπΉ)/(π΄πππ ππ β π΄π΅πΆ)=1/4

Example 5
Important

Example 8 Important

Example 10 Important

Example 14 Important

Theorem 6.1 - Basic Proportionality Theorem (BPT) Important

Theorem 6.7 Important

Ex 6.2, 4 Important

Ex 6.2, 5 Important

Ex 6.2, 6 Important

Ex 6.2, 9 Important

Ex 6.3, 11 Important

Ex 6.3, 12 Important

Ex 6.3, 13 Important

Ex 6.3, 14 Important

Ex 6.3, 15 Important

Ex 6.4, 1 Important

Ex 6.4, 3 Important

Ex 6.4, 5 Important You are here

Ex 6.5, 2 Important

Ex 6.5, 3 Important

Ex 6.5, 8 Important

Ex 6.5, 11 Important

Ex 6.5, 12 Important

Ex 6.5, 15 Important

Class 10

Important Questions for Exam - Class 10

- Chapter 1 Class 10 Real Numbers
- Chapter 2 Class 10 Polynomials
- Chapter 3 Class 10 Pair of Linear Equations in Two Variables
- Chapter 4 Class 10 Quadratic Equations
- Chapter 5 Class 10 Arithmetic Progressions
- Chapter 6 Class 10 Triangles
- Chapter 7 Class 10 Coordinate Geometry
- Chapter 8 Class 10 Introduction to Trignometry
- Chapter 9 Class 10 Some Applications of Trignometry
- Chapter 10 Class 10 Circles
- Chapter 11 Class 10 Constructions
- Chapter 12 Class 10 Areas related to Circles
- Chapter 13 Class 10 Surface Areas and Volumes
- Chapter 14 Class 10 Statistics
- Chapter 15 Class 10 Probability

About the Author

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 8 years. He provides courses for Maths and Science at Teachoo. You can check his NCERT Solutions from Class 6 to 12.