Intersecting Secants Theorem

Example :

In the circle shown, if $\mathit{\text{M}}\mathit{\text{N}}=10,\mathit{\text{N}}\mathit{\text{O}}=17,\mathit{\text{M}}\mathit{\text{P}}=9$ , then find the length of $\mathit{\text{P}}\mathit{\text{Q}}$ .

Substitute.

$\begin{array}{l}\mathit{\text{M}}\mathit{\text{N}}\cdot \mathit{\text{M}}\mathit{\text{O}}=\mathit{\text{M}}\mathit{\text{P}}\cdot \mathit{\text{M}}\mathit{\text{Q}}\\ \left(\mathit{\text{M}}\mathit{\text{N}}\right)\left(\mathit{\text{M}}\mathit{\text{N}}+\mathit{\text{N}}\mathit{\text{O}}\right)=\left(\mathit{\text{M}}\mathit{\text{P}}\right)\left(\mathit{\text{M}}\mathit{\text{P}}+\mathit{\text{P}}\mathit{\text{Q}}\right)\\ \left(10\right)\left(10+17\right)=\left(9\right)\left(9+\mathit{\text{P}}\mathit{\text{Q}}\right)\\ \left(10\right)\left(27\right)=\left(9\right)\left(9+\mathit{\text{P}}\mathit{\text{Q}}\right)\\ 270=81+9\mathit{\text{P}}\mathit{\text{Q}}\\ 189=9\mathit{\text{P}}\mathit{\text{Q}}\\ 21=\mathit{\text{P}}\mathit{\text{Q}}\end{array}$

Therefore, $\mathit{\text{P}}\mathit{\text{Q}}=21$ units.