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Tangent and Secant

In the figure...

Question

In the figure given below, diameter AB and chord CD of a circle meet at P. PT is a tangent to the circle at T. CD=7.8 cm, PD = 5 cm, PB = 4 cm. Select the statements that are true.

AB = 9 cm

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AB = 12 cm

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PT = 2 cm

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PT = 8 cm

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Solution

The correct option is B
AB = 12 cm

PT is tangent and PDC is secant out to the circle
$∴ P T^{2} = P C \times P D$
$P T^{2} = (5 + 7.8) \times 5 = 12.8 \times 5$
$P T^{2} = 64 \Rightarrow P T = 8 cm$
In $Δ O T P$
$P T^{2} + O T^{2} = O P^{2}$
$8^{2} + x^{2} = (x + 4)^{2} \Rightarrow 64 + x^{2} = x^{2} + 16 + 8 x$
$64 - 16 = 8 x \Rightarrow 48 = 8 x$
$x = \frac{48}{8} = 6 c m$
$∴ Radius = 6 c m$
$A B = 2 \times 6 = 12 c m$

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In the given figure, PT is tangent to the circle at T. If PA = 4 cm and AB = 5 cm, find PT. AB is equal to diameter.

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