This is a geometry problem involving a triangle. From the diagram:

1. There is a large triangle, and within it, a smaller triangle that appears to be similar to the larger triangle.
2. The dimensions of the larger triangle are labeled: the height is $32$, the base is $24 + 9 = 33$.
3. The smaller triangle's height is unknown ($x$), and its base is $9$.

Since the two triangles are similar, their corresponding sides are proportional. Let's solve for $x$:

$\frac{\text{Height of smaller triangle}}{\text{Height of larger triangle}} = \frac{\text{Base of smaller triangle}}{\text{Base of larger triangle}}$

Substitute the given values: $\frac{x}{32} = \frac{9}{33}$

Solve for $x$: $x = \frac{9}{33} \cdot 32$

Simplify: $x = \frac{9 \cdot 32}{33} = \frac{288}{33} \approx 8.73$

Thus, the height of the smaller triangle is approximately $8.73$.

Would you like a detailed breakdown of the steps or have any additional questions?

5 Related Questions:

1. How can we verify that the two triangles are similar?
2. What is the mathematical definition of similar triangles?
3. How do we calculate proportions in other geometric shapes?
4. What happens if the base of the smaller triangle changes to $12$? How does $x$ change?
5. Can this concept be applied to real-world measurements, like scaling models?

Tip:

Always confirm the similarity of triangles by checking angles or proportional sides before applying proportional reasoning.