Question

P is the centre of the circle . Prove that $\angle XPZ=2(\angle XZY+\angle YXZ)$. [3 MARKS]

Answer 1

Concept: 1 Mark
Steps: 1 Mark
Answer: 1 Mark

Given: A circle with centre P and XY, YZ are the two chords.

To prove:
$\angle XPZ=2(\angle XZY+\angle YXZ)$

Proof: In a circle with centre P, arc XY subtends $\angle XPY$ at the centre and $\angle XZY$ at remaining part of the circle.

$\angle XPY=2\angle XZY$ .......(1)

Similarly, arc YZ subtends $\angle YPZ$ at the centre and $\angle YXZ$ at remaining part

$\angle YPZ=2\angle YXZ$ ........(2)

Adding (1), and (2), we get

$\angle XPY+\angle YPZ=2\angle XZY+2\angle YXZ$

$\angle XPZ=2(\angle XZY+\angle YXZ)$