CPM Homework Help : CCG Problem 10-54

In the diagram at right, $⊙M$ has radius $14$ feet and $⊙A$ has radius $8$ feet. $\overleftrightarrow{ER}$ is tangent to both $⊙M$ and $⊙A$ . If $NC=17$ feet, find $ER$ .

Two circles, left with center M, and radius, M, E, right with center A, and radius, A, r. Line E R is tangent to both circles, intersecting left circle at, E, and right circle at, r. Centers M and A are connected by a line segment which intersects left circle at, N, and right circle at, c. A dashed segment from, A, perpendicular to segment, M, E, at point, b.

Hint 1:

Think about how the radius of $⊙M$ will affect $\overline{MN}$ .
How will the radius of $⊙A$ affect $\overline{AC}$ ?

Hint 2:

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

Step 1:

$MA = 14 + 17 + 8\\ MA = 39$

More Help:

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

Step 2:

$ME = MB + BE\\ \;\;\; 14 = MB + 8\\ MB = 6$

Step 3:

Use Pythagorean Theorem to find the length of the last side.
$6^2 + (BA)^2 = 39^2\\ 36 + (BA)^2 = 1521\\ \qquad \, (BA)^2 = 1485\\ \qquad \quad \;\, BA \approx 38.535$

Hint 3:

$BA=ER$

Answer:

$ER\approx38.5\;\text{ft}$