Question

Examine the diagram, where AB and CD are secants to the circle, and they intersect inside of the circle at point P, which is not the center.

Answer 1

Step 1

State the intersecting chord theorem according to the given question

$\begin{gathered} AP\times PB=CP\times PD \\ AP=21 \\ PB=2x \\ CP=x+5 \\ PD=27 \end{gathered}$

Step 2

Substitute

$21(2x)=27(x+5)_{}$$\begin{gathered} 42x=27x+135 \\ 42x-27x=135 \\ 15x=135 \\ \frac{15x}{15}=\frac{135}{15} \\ x=9 \end{gathered}$

Hence, x=9