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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 is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths and Science at Teachoo.