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 = 39/10, 54/25, 64/35, 89/10

Explanation:

Step1: Expand the right - hand side

Expand \((r + 5)^2\) using the formula \((a + b)^2=a^{2}+2ab + b^{2}\). So \((r + 5)^2=r^{2}+10r + 25\), and the equation \(r^{2}+8^{2}=(r + 5)^2\) becomes \(r^{2}+64=r^{2}+10r + 25\).

Step2: Simplify the equation

Subtract \(r^{2}\) from both sides of the equation \(r^{2}+64=r^{2}+10r + 25\). We get \(64=10r + 25\).

Step3: Solve for \(r\)

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

Answer:

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