Question

a circle with center o is inscribed in a triangle PQR having area 189 cm^2. if the circle touches QR =12cm and TR=9 cm .Find PQ AND PR.

Answer 1

See figure:

Given,
OS = OT = OU = 6 cm (radii of circle)
QT = 12 cm
TR = 9 cm
QR = QT + TR = 12+9cm = 21 cm
∴ QT = QS = 12 cm
TR = RU = x cm
∵ PQ = PS + SQ = (12+x) cm
∴ PR = PU + RU = (9+x) cm

We know,
ar (ΔOQR) + ar (ΔOPQ) = ar (ΔPQR)
⇒1/2 × QR × OT + 1/2 × PR × OU + 1/2 × PQ × OS = 189 cm²
⇒1/2 × (12+x)×6+1/2×(9+x)×6+1/2×21×6=189
⇒1/2×(12+x+9+x+21) = 189
⇒3(42+2x)=189
⇒42+2x=63
⇒2x=21
⇒x=21/2 = 10.5
∴PQ = (12+10.5) = 22.5 cm
∴PR = (9+10.5) = 19.5 cm