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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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Ex 6.4, 5 D, E and F are respectively the mid-points of sides AB, BC and CA of Ξ”ABC. Find the ratio of the areas of Ξ”DEF and Ξ”ABC. Given: Ξ” ABC & D,E,F mid-points of AB,BC & CA respectively To find: (π‘Žπ‘Ÿ βˆ†π·πΈπΉ)/(π‘Žπ‘Ÿ βˆ†π΄π΅πΆ) Note: Since we need to find ratio of area of Ξ”DEF and Ξ”ABC. 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 Ξ”ABC , D and F are mid-points of AB and AC resp., ∴ DF βˆ₯ BC So, DF βˆ₯ BE also Similarly, E and F are mid-points of BC and AC resp. EF βˆ₯ AB Hence, EF βˆ₯ DB From (1) & (2) DF βˆ₯ BE & FE βˆ₯ 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 Ξ”EDF and Ξ”ABC ∠ DFE =∠ABC ∠ EDF= ∠ ACB By using AA similarity criterion Ξ” DEF ∼ 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 ∴ (π‘Žπ‘Ÿ βˆ†π·πΈπΉ)/(π‘Žπ‘Ÿ βˆ†π΄π΅πΆ) = 𝐷𝐸2/𝐴𝐢2 (π‘Žπ‘Ÿ βˆ†π·πΈπΉ)/(π‘Žπ‘Ÿ βˆ†π΄π΅πΆ) = 𝐹𝐢2/𝐴𝐢2 (π΄π‘Ÿπ‘’π‘Ž π‘œπ‘“ βˆ†π·πΈπΉ)/(π΄π‘Ÿπ‘’π‘Ž π‘œπ‘“ βˆ†π΄π΅πΆ)=( 𝐴𝐢/2 )^2/(𝐴𝐢)2 (π΄π‘Ÿπ‘’π‘Ž π‘œπ‘“ βˆ†π·πΈπΉ)/(π΄π‘Ÿπ‘’π‘Ž π‘œπ‘“ βˆ†π΄π΅πΆ)=((𝐴𝐢)2/4)/(𝐴𝐢)2 (π΄π‘Ÿπ‘’π‘Ž π‘œπ‘“ βˆ†π·πΈπΉ)/(π΄π‘Ÿπ‘’π‘Ž π‘œπ‘“ βˆ†π΄π΅πΆ)=(1/4)/1 Hence , (π΄π‘Ÿπ‘’π‘Ž π‘œπ‘“ βˆ† 𝐷𝐸𝐹)/(π΄π‘Ÿπ‘Žπ‘’ π‘œπ‘“ βˆ† 𝐴𝐡𝐢)=1/4

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.