point b is a point of tangency. find the radius r of ⊙c.

Answer:

18

Explanation:

Step1: Apply tangent - radius property

Since point $B$ is a point of tangency, $\angle ABC = 90^{\circ}$, and $\triangle ABC$ is a right - triangle.

Step2: Use the Pythagorean theorem

In right - triangle $\triangle ABC$, by the Pythagorean theorem $AC^{2}=AB^{2}+BC^{2}$. Here, $AC=r + 16$, $AB = 24$, and $BC=r$. So, $(r + 16)^{2}=24^{2}+r^{2}$.

Step3: Expand the left - hand side

Expand $(r + 16)^{2}$: $r^{2}+32r+256=576+r^{2}$.

Step4: Simplify the equation

Subtract $r^{2}$ from both sides of the equation: $32r+256 = 576$.

Step5: Solve for $r$

Subtract 256 from both sides: $32r=576 - 256=320$. Then, divide both sides by 32: $r=\frac{320}{32}=18$.