AB is a diameter of a circle. The length of AB=5cm. If O is the centre of the circle and the length of tangent segment BT=12cm , determime CT ?

Since AB is the diameter and PB is the tangent.

So, AB ⊥ PB (Radius of the circle is perpendicular to the tangent through point of contacy)

⇒ ABC is a right triangle right angled at B

⇒ AT^{2} = AB^{2} + BT^{2} = (5 cm)^{2} + (12 cm)^{2} = 25 cm^{2} + 144 cm^{2} = 169 cm^{2} (Pythagoras theorem)

⇒ AT = 13 cm

Now, we know that

If PAB is a secant to a circle intersecting the circle to** **A and B and PT is a tangent, then PA × PB = PT^{2}

Here is the link for the same.

So,

BT^{2} = CT × AT

⇒ 144 cm^{2} = CT × 13 cm

⇒ CT = 144/13 cm

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