We want to find the length of the line segment $\overline{J M} .$

In the diagram we can see that $G H,$ $J K,$ and $M N$ intersect the transversal $H N .$ Moreover, the angles they form — $∠ H,$ $∠ K,$ and $∠ N$ — are corresponding and congruent. Therefore, $G H,$ $J K,$ and $M N$ are parallel lines. Let's recall the Three Parallel Lines Theorem.

Three Parallel Lines Theorem
If three parallel lines intersect two transversals, then they divide the transversals proportionally.

With this logic, we know that

G H,

J K,

and

M N

divide

G M

and

H N

proportionally.

\frac{H K}{K N} = \frac{G J}{J M}

Since we know that

H K = 15,

K N = 18,

and

G J = 16,

we can substitute these values into the above equation and solve for

J M .

\frac{H K}{K N} = \frac{G J}{J M}

Substitute values

\frac{1 5}{1 8} = \frac{1 6}{J M}

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Solve for

J M

$LHS \cdot J M = RHS \cdot J M$

\frac{1 5}{1 8} \cdot J M = 16

$LHS \cdot \frac{1 8}{1 5} = RHS \cdot \frac{1 8}{1 5}$

J M = 16 \cdot \frac{1 8}{1 5}

$a \cdot \frac{b}{c} = \frac{a \cdot b}{c}$

J M = \frac{2 8 8}{1 5}

Calculate quotient

J M = 19.2

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