Question

The pair of figures to the right are similar. The area of one figure is given. Find the area of the other figure to the nearest whole number.
Area of larger triangleequals165 ft squared
25 ft
30 ft



One larger triangle is to the left of a smaller triangle. The two triangles have the same shape. One edge of the larger triangle is labeled "30 feet" and the corresponding side of the smaller triangle is labeled "25 feet".
Question content area bottom
Part 1
The area of the smaller triangle is
  
enter your response here ft squared. ​(Round to the nearest whole number as​ needed.)

Answer 1

To find the area of the smaller triangle when the two triangles are similar, we can use the properties of similar figures. The ratio of the areas of similar figures is equal to the square of the ratio of their corresponding side lengths.

Given:
- Area of the larger triangle = 165 ft²
- Corresponding sides: larger = 30 ft, smaller = 25 ft

First, we calculate the ratio of the sides:

\[
\text{Ratio} = \frac{\text{Smaller Side}}{\text{Larger Side}} = \frac{25}{30} = \frac{5}{6}
\]

Next, we find the ratio of the areas:

\[
\left(\frac{5}{6}\right)^2 = \frac{25}{36}
\]

Now, we use the ratio of the areas to find the area of the smaller triangle:

Let \( A_s \) be the area of the smaller triangle. We can set up the following proportion based on the areas:

\[
\frac{A_s}{165} = \frac{25}{36}
\]

Now, solve for \( A_s \):

\[
A_s = 165 \times \frac{25}{36}
\]

Calculating \( A_s \):

\[
A_s = 165 \times \frac{25}{36} = \frac{4125}{36} = 114.5833 \approx 115 \text{ ft}^2
\]

Thus, the area of the smaller triangle is approximately **115 ft²** when rounded to the nearest whole number.