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prove that the ratio of the square of two similar triangles is equal to the ratio of the squares of their corresponding sides

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Ramesh answered 3 year(s) ago

Ratio of the areas of two similar triangles.

Prove that the ratio of the areas of two similar triangles is equal to the ratio of the squares of any two corresponding angle bisector segments.

Class-X Maths

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Asked by Devanshi

Dec 28

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Ramesh , SubjectMatterExpert

Member since Apr 01 2014

Consider two triangles ABC and DEF.

AX and DY are the bisectors of the angles A and D respectively.



Ratio of the areas of two similar triangles is equal to the ratio of the squares of any two corresponding sides. so,

Area (ΔABC) / Area (ΔDEF) = AB2/ DE2 -----(1)

ΔABC ~ ΔDEF ⇒ ∠A = ∠D

1/ 2 ∠A = 1 / 2 ∠D ⇒ ∠BAX = ∠EDY

Consider ΔABX and ΔEDY

∠BAX = ∠EDY

∠B = ∠E

So, ΔABX ~ ΔEDY [By A-A Similarity]

AB/DE = AX/DY

⇒ AB2/DE2 = AX2/DY2 --------- (2)

From equations (1) and (2), we get

Area (ΔABC) / Area (ΔDEF) = AX2/ DY2

Hence proved.