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Prove that the area of the equilateral triangle drawn on the hypotenuse of a right angled triangle is equal to the sum of the areas of the equilateral triangle drawn on the other two sides of the triangle.

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Let a right triangle QPR in which ÐP is right angle and PR = y, PQ = x.

Three equilateral triangles ∆PER, ∆PFR and ∆RQD are drawn on the three sides of ∆PQR.

Again, let area of triangles made on PR, PQ are A1, A2 and A3, respectively.

To prove A3 = A1 + A2

In ∆RPQ,

By Pythagoras theorem,

QR2 = PR2 + PQ2

⇒ QR2 = y2 + x2

We know that,

A1 + A2

[from Equation (i) and (ii)]

Hence proved.

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