Question

This is a practice assessment I just need to get it done to understand how to do it better!

Answer 1

Since P is a point of tangency of the circumference O, then the angle OPQ is a right angle.

Then, the triangle OPQ is a right triangle. Use the Pythagorean Theorem to write an equation for r. Then, solve that equation.

Since OQ is the hypotenuse of OPQ, then:

$OP^2+PQ^2=OQ^2$

Since OP=r, OQ=r+9 and PQ=15, then:

$r^2+15^2=(r+9)^2$

Expand the quadratic binomial on the right member of the equation:

$\Rightarrow r^2+15^2=r^2+2\cdot r\cdot9+9^2$

Since the term r^2 appears on both sides, simplify the equation:

$\Rightarrow15^2=2\cdot r\cdot9+9^2$

Simplify all terms and solve for r:

$\begin{gathered} \Rightarrow225=18r+81 \\ \Rightarrow225-81=18r \\ \Rightarrow144=18r \\ \Rightarrow\frac{144}{18}=r \\ \Rightarrow8=r \end{gathered}$

Therefore, the radius of the circumference is:

$8$