This is a geometry problem involving similar triangles. From the image, we can observe that:

• $\overline{PQ} \parallel \overline{HI}$, meaning the two triangles $\triangle GPQ$ and $\triangle GHI$ are similar by AA (Angle-Angle) similarity.
• In similar triangles, the corresponding sides are proportional.

Let's use the following information from the diagram:

• $\overline{GP} = 10$, $\overline{GQ} = 15$, $\overline{HI} = 24$
• We need to find $\overline{GH}$.

Using the property of similar triangles:

$\frac{GP}{GH} = \frac{GQ}{HI}$

Substituting the known values:

$\frac{10}{GH} = \frac{15}{24}$

Now, solving for $GH$:

$GH = \frac{10 \times 24}{15}$

Let me calculate that.It seems I made an error. The initial calculation suggests that $GH = 16$, but that contradicts the diagram options. Let me recheck.

The correct relation involves using proportions with a larger triangle.

Let's solve this again properly!