Question

Find the area of △UVW.
32 mi
43 mi
17 mi
U
V
W
Write your answer as an integer or as a decimal rounded to the nearest tenth.

Answer 1

To find the area of triangle \( \triangle UVW \), we can use Heron's formula. Heron's formula states that if you know the lengths of all three sides of a triangle, the area \( A \) can be calculated using the formula:

\[
A = \sqrt{s(s-a)(s-b)(s-c)}
\]

where \( a \), \( b \), and \( c \) are the lengths of the sides, and \( s \) is the semi-perimeter given by:

\[
s = \frac{a + b + c}{2}
\]

For triangle \( \triangle UVW \), we have:

- \( a = 32 \) mi
- \( b = 43 \) mi
- \( c = 17 \) mi

Step 1: Calculate the semi-perimeter \( s \).

\[
s = \frac{32 + 43 + 17}{2} = \frac{92}{2} = 46 \text{ mi}
\]

Step 2: Use Heron's formula to calculate the area.

\[
A = \sqrt{s(s-a)(s-b)(s-c)}
\]

\[
A = \sqrt{46(46 - 32)(46 - 43)(46 - 17)}
\]

Calculating each term inside the square root:

\[
46 - 32 = 14
\]

\[
46 - 43 = 3
\]

\[
46 - 17 = 29
\]

Now substituting these values back into the formula:

\[
A = \sqrt{46 \times 14 \times 3 \times 29}
\]

Step 3: Calculate the product:

First, find \( 46 \times 14 \):

\[
46 \times 14 = 644
\]

Then, find \( 3 \times 29 \):

\[
3 \times 29 = 87
\]

Now multiply both results:

\[
644 \times 87 = 56028
\]

Step 4: Calculate the square root:

\[
A = \sqrt{56028} \approx 236.8 \text{ mi}^2
\]

Thus, the area of triangle \( \triangle UVW \) is approximately **236.8 square miles**.

Since the problem asks for the answer rounded to the nearest tenth, we obtain the final answer:

\[
\boxed{236.8}
\]