  ### Home > CCG > Chapter 10 > Lesson 10.1.5 > Problem10-54

10-54.

In the diagram below, $⊙M$ has radius $14$ feet and $⊙A$ has radius $8$ feet. $\overleftrightarrow{ E R }$ is tangent to both $⊙M$ and $⊙A$. If $\overline{NC} = 17$ feet, find $\overline{ER}$. Homework Help ✎ Think about how the radius of $⊙M$ will affect $\overline{MN}$.
How will the radius of $⊙A$ affect $\overline{AC}$?

After finding the lengths of $\overline{MN}$ and $\overline{AC}$, find lengths of the remaining sides.
$\overline{MA} = \overline{MN} + \overline{NC} + \overline{CA}$
$\overline{ME} = \overline{MN}$

$\overline{BE} ≅ \overline{AR}$ because $\overline{ME}$ and $\overline{AR} ⊥ \overline{ER}$ and by the definition of a tangent.

$\overline{BA} = \overline{ER}$

$\overline{MA} = 14 + 17 + 8$
$\overline{MA} = 39$

$\overline{ME} = \overline{MB} + \overline{BE}$
$14 = \overline{MB} + 8$
$\overline{MB} = 6$

Use Pythagorean Theorem to find the last side.
$6^2 + \overline{BA^2} = 392$
$36 + \overline{BA^2} = 1521$
$\overline{BA^2} = 1485$
$\overline{BA} = 38.535$

$\overline{ER} = 38.5$ ft.