Math Snap

STEP 1

What is this asking?
We need to find the area of a triangle given the lengths of its three sides.
Watch out!
We can't just multiply the sides together!
We need a formula that relates the area of a triangle to its side lengths.

STEP 2

1. Calculate the semi-perimeter.
2. Apply Heron's formula.

STEP 3

Alright, first things first, we need the semi-perimeter, which is half the perimeter.
Let's call the sides $a$, $b$, and $c$.
We have $a =$ 35 mi, $b =$ 36 mi, and $c =$ 29 mi.

STEP 4

The perimeter is just the sum of the sides:
$$a + b + c = 35 + 36 + 29 = \mathbf{100} \text{ mi}.$$

STEP 5

Now, the semi-perimeter, which we'll call $s$, is half of that:
$$s = \frac{100}{2} = \mathbf{50} \text{ mi}.$$ So our semi-perimeter is 50 mi.

STEP 6

Now for the main event: Heron's formula!
It says the area of a triangle is
$$\sqrt{s(s-a)(s-b)(s-c)}.$$ Remember, $s$ is the semi-perimeter, and $a$, $b$, and $c$ are the side lengths.

STEP 7

Let's plug in our values.
We have $s =$ 50, $a =$ 35, $b =$ 36, and $c =$ 29.
So we get:
$$\sqrt{50(50-35)(50-36)(50-29)}$$ $$= \sqrt{50 \cdot 15 \cdot 14 \cdot 21}$$$$= \sqrt{220500}.$$

STEP 8

Calculating the square root gives us approximately 469.57 mi$^2$.
Rounding to the nearest tenth gives us 469.6 mi$^2$.

The area of triangle WXY is approximately 469.6 mi$^2$.