# Problems on Relation Between Tangent and Secant

Here we will solve different types of Problems on relation between tangent and secant.

1. XP is a secant and PT is a tangent to a circle. If PT = 15 cm and XY = 8YP, find XP.

Solution:

XP = XY + YP = 8YP + YP = 9YP.

Let YP = x. Then XP = 9x.

Now, XP × YP = PT2, as the product of the segments of a secant is equal to the square of the tangent.

Therefore, 9x ∙ x = 152 cm2

⟹ 9x2 = 152 cm2

⟹ 9x2 = 225 cm2

⟹ x2 = $$\frac{225}{9}$$ cm2

⟹ x2 = 25 cm2

⟹ x = 5 cm.

Therefore, XP = 9x = 9 ∙ 5 cm = 45 cm.

2. XYZ is an isosceles triangle in which XY = XZ. If N is the mid point of XZ, prove that XY = 4 XM.

Solution:

Let XY = XZ = 2x.

Then XN = $$\frac{1}{2}$$XZ = x.

XY is a secant and XN is a tangent.

Therefore, XM × XY = XN2 (Product of segments of secant = square of tangent).

Therefore, XM × 2x = x2

⟹ XM = $$\frac{x}{2}$$.

Therefore, XY = 2x = 4 ∙ $$\frac{x}{2}$$ = 4XM

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