Question

determining the radius
\\(\overline{cb}\\) is tangent to \\(\odot a\\) at point \\(c\\). find the radius.
\\(\overline{cb} \perp \overline{ac}\\) by the radius-tangent theorem, so \\(\angle c\\) is a right angle.
\\(\triangle 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 options: 39/10, 54/25, 64/35, 89/10

Explanation:

Step1: Simplify the equation

We have the equation \( r^{2}+8^{2}=(r + 5)^{2}\), which expands to \(r^{2}+64=r^{2}+10r + 25\). Subtract \(r^{2}\) from both sides:
\(r^{2}-r^{2}+64=r^{2}-r^{2}+10r + 25\)
\(64 = 10r+25\)

Step2: Solve for r

Subtract 25 from both sides:
\(64 - 25=10r+25 - 25\)
\(39 = 10r\)

Then divide both sides by 10:
\(r=\frac{39}{10}\)

Answer:

\(\frac{39}{10}\) (or 39/10 from the options)