Question

Example 11: If APQR is an isosceles triangle such that PO = PR, then prove that the altitude
PS from P on QR, bisects OR (Fig. 9.35).
P
R
S
Fig. 9.35​

Answer 1

In the figure, ∆PQR is an isosceles PQ = PR

∠PQR = 35°

∴ ∠PRQ = 35°

But ∠PQR + ∠PRQ + ∠QPR = 180° (Sum of angles of a triangle)

⇒ 35° + 35° + ∠QPR = 180°

⇒ 70° + ∠QPR = 180°

∴ ∠QPR = 180° – 70° = 110°

∵ ∠QSR = ∠QPR (Angle in the same segment of circles)

∴ ∠QSR = 110°

But PQTR is a cyclic quadrilateral

∴ ∠QTR + ∠QPR = 180°

⇒ ∠QTR + 110° = 180°

⇒ ∠QTR = 180° -110° = 70°

Hence ∠QTR = 70°