We are given that $\overline{ED}$ and $\overline{EF}$ are tangent to circle $C$ at points $D$ and $F$ respectively. The length of $\overline{ED}$ is $9x+12$ and the length of $\overline{EF}$ is $15x-24$. We are asked to find the value of $x$ and the length of $\overline{ED}$.

1. Problem Description

We are given that $\overline{ED}$ and $\overline{EF}$ are tangent to circle $C$ at points $D$ and $F$ respectively. The length of $\overline{ED}$ is $9x+12$ and the length of $\overline{EF}$ is $15x-24$. We are asked to find the value of $x$ and the length of $\overline{ED}$.

2. Solution Steps

If two segments are tangent to a circle from the same external point, then the segments are congruent. In this case, $\overline{ED}$ and $\overline{EF}$ are tangent to circle $C$ from point $E$. Therefore, $ED = EF$.

$9x + 12 = 15x - 24$

Subtract $9x$ from both sides:

$12 = 6x - 24$

Add 24 to both sides:

$36 = 6x$

Divide both sides by 6:

$x = 6$

Now we can find the length of $\overline{ED}$ by substituting $x = 6$ into the expression for $ED$:

$ED = 9x + 12 = 9(6) + 12 = 54 + 12 = 66$

3. Final Answer

$x = 6$

$ED = 66$