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.
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