Question

In a right-angled triangle PQR right-angled at Q, QR = x cm, PQ = (x + 7) cm and area = 30 cm². Find the sides of the triangle.

Answer 1

Area of right-angled ΔPQR = `(1)/(2) xx "QR" xx "PQ"`

⇒ 30 = `(1)/(2) xx x xx (x + 7)`

⇒ 60 = x² + 7x
⇒ x² + 7x - 60 = 0
⇒ x² + 12x - 5x - 60 = 0
⇒ x(x + 12) -5(x + 12) = 0
⇒ (x + 12)(x - 5) = 0
⇒ x + 12 = 0 x - 5 = 0
⇒ x = -12 or x = 5
But, length of a side cannot be negatice.
Hence,
x = 5cm = QR
x + 7 
= 5 + 7
= 12cm
= PQ
In right-angled ΔPQR, by Pythagoras theorem,
PR² = PQ² + QR²
= 12² + 5²
= 144 + 25
= 169
⇒ PR = 13cm
Hence, PQ = 12cm, QR = 5cm and PR = 13cm.