Question

cb is tangent to ⊙a at point c. find the radius.
cb ⊥ ac by the radius-tangent theorem, so ∠c is a right angle.
△abc is a right triangle, so apply the pythagorean theorem.
use the steps and solve for the radius.

1. ( r^2 + 8^2 = (r + 5)^2 )
2. ( r^2 + 64 = r^2 + 10r + 25 )

r = dropdown with 39/10, 54/25, 64/35, 89/10

Explanation:

Step1: Simplify the equation

We have the equation \( r^{2}+64 = r^{2}+10r + 25 \). Subtract \( r^{2} \) from both sides of the equation.
\( r^{2}-r^{2}+64=r^{2}-r^{2}+10r + 25 \)
Which simplifies to \( 64 = 10r+25 \).

Step2: Solve for r

Subtract 25 from both sides: \( 64 - 25=10r+25 - 25 \), so \( 39 = 10r \). Then divide both sides by 10: \( r=\frac{39}{10} \).

Answer:

\(\frac{39}{10}\)