In the first picture below, the line joining two points on a circle is extended by 4 centimetres and a tangent is drawn from this point. Its length is 6 centimetres, as shown:
The second picture shows the same line extended by 1 centimetre more and a tangent drawn from this point. What is the length of this tangent?
Let us label the diagram
We know that,
The product of an intersecting line and its part outside the circle is equal to the square of tangent.
i.e.
AB × AP = CP2
In first case,
AP = 4 cm
CP = 6 cm
⇒ AB × 4 = 62
⇒ 4AB = 36
⇒ AB = 9 cm
In second case,
AP = 5 cm
To find : CP
As, length of AB is not changed we have,
AB × AP = CP2
⇒ CP2 = 5(9)
⇒ CP2 = 45
⇒ CP = 3√5 cm