# Problems on Properties of Tangents

Here we will solve different types of Problems on properties of tangents.

1. A tangent, PQ, to a circle touches it at Y. XY is a chord such that ∠XYQ = 65°. Find ∠XOY, where O is the centre of the circle.

Solution:

Let Z be any point on the circumference in the segment alternate ∠XYQ.

Therefore, ∠XZY = ∠XYQ = 65°, as the angle between a chord and a tangent is equal to the angle in the alternate segment.

∠XOY = 2∠XZY, as angle at the centre is double the angle at the circumference.

Therefore, ∠XOY = 2 × 65° = 130°.

2. XY is a chord of a given circle, which on producing, meets the tangent TZ at Z. If XY = 5 cm and YZ = 4 cm, find TZ.

Solution:

XZ = XY + YZ

= 5 cm + 4 cm

= 9 cm.

Again, we know that

XZ × YZ = TZ^2

⟹ 9 cm × 4 cm = TZ^2

⟹ 36 cm^2 = TZ^2

⟹ TZ = $$\sqrt{36 cm^2}$$

⟹ TZ = 6 cm.