Question

Find the area of triangle VWX
X= 49 km
W= 22 km
V= 35 km
write your answer as an integer or as a decimal rounded to the nearest tenth
_____ km ^2

Answer 1

To find the area of triangle VWX with sides \( X = 49 \) km, \( W = 22 \) km, and \( V = 35 \) km, we can use Heron's formula. First, we need to calculate the semi-perimeter \( s \) of the triangle.

The semi-perimeter \( s \) is given by:

\[
s = \frac{X + W + V}{2}
\]

Substituting the values:

\[
s = \frac{49 \text{ km} + 22 \text{ km} + 35 \text{ km}}{2} = \frac{106 \text{ km}}{2} = 53 \text{ km}
\]

Now we apply Heron's formula for the area \( A \):

\[
A = \sqrt{s(s - X)(s - W)(s - V)}
\]

Calculating each term:

\[
s - X = 53 \text{ km} - 49 \text{ km} = 4 \text{ km}
\]
\[
s - W = 53 \text{ km} - 22 \text{ km} = 31 \text{ km}
\]
\[
s - V = 53 \text{ km} - 35 \text{ km} = 18 \text{ km}
\]

Now substituting into Heron's formula:

\[
A = \sqrt{53 \text{ km} \times 4 \text{ km} \times 31 \text{ km} \times 18 \text{ km}}
\]

Calculating the product:

\[
A = \sqrt{53 \times 4 \times 31 \times 18}
\]

Now calculating \( 53 \times 4 = 212 \).

Next, \( 31 \times 18 = 558 \).

Now we multiply these two results:

\[
212 \times 558 = 118776
\]

Finally, we find the square root:

\[
A = \sqrt{118776} \approx 344.5 \text{ km}^2
\]

Thus, the area of triangle VWX is approximately \( 344.5 \) km². Hence, rounded to the nearest tenth, the area is:

\[
\boxed{344.5}
\] km².