Question

Equilateral triangles are drawn on the sides of a right triangle prove that area of the right triangle drawn on the hypotenuse is equal to the sum of the areas of triangles drawn on the other two sides

Answer 1

Given:
ABCis a right ∆. Angle ABC= 90°.AC is hypotenuse. Equilateral triangles PAB, QBC,RCA are drawn on the sides AB, BC, CA.

To Prove:
ar(∆PAB)+ar(∆QBC)= ar(∆RCA)

Proof & figure is in the attachment.
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Hope this will help you....

Answer 2

prove : triangle PAB and triangle QBC and triangle RCR similar since all the triangle are equilateral

Two triangle PNB similar triangle QBC similar triangle RCA

Hence,

Area of triangle PAV / area of triangle are RCA.= ab square upon ac square --------------(1)

Area of triangle ABC / area of triangle are = BC square + AC square

------------(2)

Adding equation (1) and (2)

area of triangle PAB / area of triangle RCA + area of triangle QBC / area of triangle RCA = AB square /AC square + BC square / AC square - AB square - BC square / AC Square

area of triangle PAB + area of triangle QBC / area of triangle RCA = AC square / AC square = 1

area of triangle PAB+ area of triangle QBC = area of triangle RCA

Hence proved

I hope it's clear for you