Isosceles Right Triangle Dimensions: Solving for Side Lengths with Area 32

In front of you is an isosceles right triangle.

The expressions listed next to the sides describe their length.

( $x>-8$ length measurements in cm).

Since the area of the triangle is 32.

Find the lengths of the sides of the triangle.

To solve this problem, we'll utilize the properties of an isosceles right triangle and the area formula:

• Step 1: Identify the expression for the legs of the triangle as $x + 8$.
• Step 2: Use the area formula for a right triangle, $\frac{1}{2} \times (\text{leg})^2 = 32$.
• Step 3: Given that the triangle is isosceles, solve for $x$ by substituting and expanding the terms.
• Step 4: Derive the hypotenuse using the known leg value and calculate $\text{leg} \times \sqrt{2}$.

Now, let's work through each step:

Step 1: Recognize both legs of the isosceles triangle are $x + 8$.

Step 2: Apply the area formula for right triangles:
We know $\frac{1}{2} \times (\text{leg}) \times (\text{leg}) = 32$. Therefore, the equation is:

Step 3: Simplify and solve for $x$.

Given the constraint $x > -8$, we discard $x + 8 = -8$ since it violates the condition. Therefore,

Recalculate $x + 8$, which states the leg is 8.

Step 4: Determine the hypotenuse.

Therefore, the side lengths of the triangle are $8, 8, 8\sqrt{2}$. Match this to the choices given, which is option 3.

The lengths of the sides of the triangle are $8, 8, 8\sqrt{2}$.