(Q22) 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.

Solution :

PQSRTUbbbaaaccc

Let ∆PQR is a right angled triangle, ∠Q = o

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

Then from Pythagoras theorem we can say a2 + 2 = c2 .....(1)

∆PSR is an equilateral triangle drawn on hypotenuse

∴ PR = PS = RS = c,

Then area of triangle on hypotenuse =
√3
 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 =
√3
4
 2.....(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'=
√3
4
 a2.....(4)

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

√
4
 a2 +
√3
4
 b2 =
√3
 [a2 + 2] =
√3
4
 c2 [∵ from eq.(1)]

= Area of equilateral triangle on the hypotenuse.

Hence Proved