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 options: 39/10, 54/25, 64/35, 89/10

Explanation:

Step1: Expand the right - hand side

Expand $(r + 5)^2$ using the formula $(a + b)^2=a^{2}+2ab + b^{2}$, where $a = r$ and $b = 5$. So $(r + 5)^2=r^{2}+10r + 25$. The equation $r^{2}+64=r^{2}+10r + 25$ is obtained from $r^{2}+8^{2}=(r + 5)^{2}$.

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: $64−25 = 10r$, which simplifies to $39 = 10r$. Then divide both sides by 10 to find $r=\frac{39}{10}$.

Answer:

$\frac{39}{10}$