Ex 6.4, 5 - D, E and F are mid-points of sides AB, BC, CA - Area of similar triangles

Ex 6.4, 5 - Chapter 6 Class 10 Triangles - Part 2
Ex 6.4, 5 - Chapter 6 Class 10 Triangles - Part 3 Ex 6.4, 5 - Chapter 6 Class 10 Triangles - Part 4

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Question 5 D, E and F are respectively the mid-points of sides AB, BC and CA of ΔABC. Find the ratio of the areas of ΔDEF and ΔABC. Given: Δ ABC & D,E,F mid-points of AB,BC & CA respectively To find: (𝑎𝑟 ∆𝐷𝐸𝐹)/(𝑎𝑟 ∆𝐴𝐵𝐶) Note: Since we need to find ratio of area of ΔDEF and ΔABC. 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 ΔABC , D and F are mid-points of AB and AC resp., ∴ DF ∥ BC So, DF ∥ BE also Similarly, E and F are mid-points of BC and AC resp. EF ∥ AB Hence, EF ∥ DB From (1) & (2) DF ∥ BE & FE ∥ 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 ∠ DFE =∠ABC Similarity, we can prove DECF is a parallelogram In a parallelogram, opposite angles are equal Hence, ∠ EDF= ∠ ACB Now , in ΔEDF and ΔABC ∠ DFE =∠ABC ∠ EDF= ∠ ACB By using AA similarity criterion Δ DEF ∼ Δ 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 ∴ (𝑎𝑟 ∆𝐷𝐸𝐹)/(𝑎𝑟 ∆𝐴𝐵𝐶) = 𝐷𝐸2/𝐴𝐶2 (𝑎𝑟 ∆𝐷𝐸𝐹)/(𝑎𝑟 ∆𝐴𝐵𝐶) = 𝐹𝐶2/𝐴𝐶2 (𝐴𝑟𝑒𝑎 𝑜𝑓 ∆𝐷𝐸𝐹)/(𝐴𝑟𝑒𝑎 𝑜𝑓 ∆𝐴𝐵𝐶)=( 𝐴𝐶/2 )^2/(𝐴𝐶)2 (𝐴𝑟𝑒𝑎 𝑜𝑓 ∆𝐷𝐸𝐹)/(𝐴𝑟𝑒𝑎 𝑜𝑓 ∆𝐴𝐵𝐶)=((𝐴𝐶)2/4)/(𝐴𝐶)2 (𝐴𝑟𝑒𝑎 𝑜𝑓 ∆𝐷𝐸𝐹)/(𝐴𝑟𝑒𝑎 𝑜𝑓 ∆𝐴𝐵𝐶)=(1/4)/1 Hence , (𝐴𝑟𝑒𝑎 𝑜𝑓 ∆ 𝐷𝐸𝐹)/(𝐴𝑟𝑎𝑒 𝑜𝑓 ∆ 𝐴𝐵𝐶)=1/4

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Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.