Question

Seg PQ is a chord of a circle. Arc PXQ is a major arc, and Arc PYQ is a minor arc. ∠POQ is the central angle of the circle:

1. Draw the figure from the given information.
2. Find the measure of ∠PXQ, ∠PYQ, and add m ∠PXQ, m ∠PYQ.
3. Find the measure of ∠POQ. Write the relation between ∠PXQ and ∠POQ.

Answer 1

(b)

Let the measure of the central angle be m ∠POQ = θ,

m ∠PXQ = `1/2 xx m ∠POQ = θ/2`,

m ∠PYQ = 180° − m ∠PXQ

= `180° - θ/2`

Sum of m ∠PXQ and m ∠PYQ:

m ∠PXQ + m ∠PYQ = `θ/2 + (180° - θ/2)`

∴ m ∠PXQ + m ∠PYQ = 180°

(c)

The measure of ∠POQ = 2 × m ∠PXQ,

And, the relation between ∠PXQ and ∠POQ is that the central angle subtended by an arc at the center is double the angle subtended by it at any point on the remaining part of the circle.