Question

In the figure below, triangle GHI is similar to triangle MHL.

Triangle G H I. Side G H is 18 and side I H is 16. Triangle M H L. Side M H is x and side H L is 24. Side G I and L M are parallel.

What is the distance between M and H?
12
16
24
27

Answer 1

To find the distance between points M and H (the length of side MH), we can use the property of similar triangles. Since triangle GHI is similar to triangle MHL, the ratios of the corresponding sides will be equal.

Given:
- Side GH (in triangle GHI) is 18.
- Side IH (in triangle GHI) is 16.
- Side HL (in triangle MHL) is 24.
- We need to find side MH (which is x).

From the information about the triangles, we set up a proportion based on the corresponding sides:

\[
\frac{GH}{MH} = \frac{IH}{HL}
\]

Substituting in the known values:

\[
\frac{18}{x} = \frac{16}{24}
\]

Now, simplify the right side:

\[
\frac{16}{24} = \frac{2}{3}
\]

So now we have:

\[
\frac{18}{x} = \frac{2}{3}
\]

Cross-multiplying gives:

\[
18 \cdot 3 = 2 \cdot x
\]

This simplifies to:

\[
54 = 2x
\]

Dividing both sides by 2, we find:

\[
x = 27
\]

Therefore, the distance between M and H is:

\[
\boxed{27}
\]