Question

If a secant and a tangent of a circle intersect in a point outside the circle, then the area of the rectangle formed by the two line segments corresponding to the secant is equal to the area of the area of the square formed by the line segment corresponding to the other tangent.

Answer 1

Given that,

$PT$ is a tangent and $PAB$ is a secent.

To prove that,

$P{T}^2=PA\times PB$

Construction : Draw seg $BT$ and $AT$

In $\Delta PAT$ and $\Delta PBT$

$\angle PTA=\angle PBT$ (Alternate segment angle)

$\angle TPA=\angle TPB$ [Common angle]

$\therefore \Delta PAT\sim \Delta PBT$ (A - A similarity)

$\frac{PT}{PB}=\frac{PA}{PT}$

$P{T}^2=PA\times PB$

Hence, proved