Question

$\overline{lk}$ is tangent to circle j at point k.
what is the length of the radius?
$\frac{6}{85}$
$\frac{85}{12}$
$\frac{121}{36}$
$\frac{157}{12}$

Explanation:

Step1: Identify right triangle property

A tangent to a circle is perpendicular to the radius at the point of tangency, so $\triangle LKJ$ is a right triangle with $\angle LKJ = 90^\circ$. The length of $LJ$ is $6 + r$.

Step2: Apply Pythagorean theorem

For right triangle $\triangle LKJ$, $LK^2 + JK^2 = LJ^2$. Substitute the known values:
$$11^2 + r^2 = (6 + r)^2$$

Step3: Expand and simplify equation

Expand the right-hand side:
$$121 + r^2 = 36 + 12r + r^2$$
Cancel $r^2$ from both sides:
$$121 = 36 + 12r$$

Step4: Solve for $r$

Subtract 36 from both sides:
$$121 - 36 = 12r$$
$$85 = 12r$$
Solve for $r$:
$$r = \frac{85}{12}$$

Answer:

$\frac{85}{12}$ (Option: $\boldsymbol{\frac{85}{12}}$)