Question

60. In the figure P is the centre prove that ZXPZ = 2 (ZXZP + ZYXZ)
Y
X
Z
Р​

Answer 1

Answer:

Arc XY subtends ∠XPY at the centre P and ∠XZY in the remaining part of the circle .

∴ ∠XPY = 2 ∠XZY ...i)

Similarly, arc YZ subtends ∠YPZ at the centre P and ∠YXZ in the remaining part of the circle.

∴ ∠YPZ = 2∠YXZ ...ii)

Adding (i) and (ii), we have

∠XPY +

Arc XY subtends ∠XPY at the centre P and ∠XZY in the remaining part of the circle .

∴ ∠XPY = 2 ∠XZY ...i)

Similarly, arc YZ subtends ∠YPZ at the centre P and ∠YXZ in the remaining part of the circle.

∴ ∠YPZ = 2∠YXZ ...ii)

Adding (i) and (ii), we have

∠XPY + ∠YPZ = 2 (∠XZY + ∠YXZ)

∠XPZ = 2 (∠XZY + ∠YXZ)