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.


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,


Area of an equilateral triangle =


Area of equilateral ∆PER,



And area of equilateral ∆PFQ,



A1 + A2


[from Equation (i) and (ii)]


Hence proved.


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