∆ABC the mid points of sides BC,CA,and AB, are D,E,F respectively.Find the ratio of ar(∆DEF) to ar(∆ABC)
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Answered by
5
the required ratio is 1 : 4
SahilKaushik:
please describe briefly
doing bro
see , 3 parallelogram will be formed with the help of congruence of ∆DEF with ∆AFE, ∆BFD, ∆DEC... so area of all triangles is equal to each other.... now sum of areas of all triangles is equal to the bigger triangle..... keep the area of all 4 triangles equal to ar(∆DEF) ,, U WILL GET RATIO 1:4
Answered by
9
Answer:
1:4
Step-by-step explanation:
Given in ΔABC, D, E and F are midpoints of sides AB, BC and CA respectively.
BC = EC
Recall that the line joining the midpoints of two sides of a triangle is parallel to third side and half of it.
Hence DF = (1/2) BC
⇒ (DF/BC) = (1/2) → (1)
Similarly, (DE/AC) = (1/2) → (2)
(EF/AB) = (1/2) → (3)
From (1), (2) and (3) we have
But if in two triangles, sides of one triangle are proportional to the sides of the other triangle, then their corresponding angles are equal and hence the two triangles are similar
Hence ΔABC ~ ΔEDF [By SSS similarity theorem]
Hence area of ΔDEF : area of ΔABC = 1 : 4
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