Question

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

Answer 1

Let △PQR is a right angled triangle, ∠Q = 90° 

Let PQ = a, QR b and PR = hypotenuse = c 

Then from Pythagoras theorem we can say 

a² + b² = c² ……… (1) 

△PSR is an equilateral triangle drawn on hypotenuse 

∴ PR = PS = RS = c, 

Then area of triangle on hypotenuse

 = \(\frac{\sqrt{3}}{4}\)c² ……… (2) 

△QRU is an equilateral triangle drawn on the side ‘QR’ = b 

∴ QR = RU = QU = b 

Then area of equilateral triangle drawn on the side = \(\frac{\sqrt{3}}{4}\)b² …….. (3) 

△PQT is an equilateral triangle drawn on another side ‘PQ’ = a 

∴ PQ = PT = QT = a 

Area of an equilateral triangle drawn an another side ‘PQ’ = \(\frac{\sqrt{3}}{4}\)a² …….. (4) 

Now sum of areas of equilateral triangles on the other two sides

= Area of equilateral triangle on the hypotenuse. 

Hence Proved.