Question

lk is tangent to circle j at point k. what is the length of the radius? 6/85 121/36 169/12 55/12

Explanation:

Step1: Apply tangent - radius property

Since $LK$ is tangent to circle $J$ at point $K$, $\angle{JKL}=90^{\circ}$, and we can use the Pythagorean theorem in right - triangle $JKL$. The distance from $L$ to the circle along the secant is $6$ and the tangent length $LK = 11$, and the radius is $r$. The length from $L$ to the center $J$ is $r + 6$.

Step2: Set up Pythagorean equation

By the Pythagorean theorem, $(r + 6)^2=r^2+11^2$. Expand the left - hand side: $r^{2}+12r + 36=r^{2}+121$.

Step3: Solve for $r$

Subtract $r^{2}$ from both sides of the equation: $12r+36 = 121$. Then subtract 36 from both sides: $12r=121 - 36=85$. Divide both sides by 12 to get $r=\frac{85}{12}$.

Answer:

$\frac{85}{12}$