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

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:

$\frac{1}{2} \times (x + 8)^2 = 32$

Step 3: Simplify and solve for $x$.

$(x + 8)^2 = 64$ $x + 8 = \sqrt{64}$ $x + 8 = 8 \quad \text{or} \quad x + 8 = -8$

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

$x + 8 = 8$ $x = 0$

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

Step 4: Determine the hypotenuse.

$hypotenuse = 8\sqrt{2}$

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 $8, 8, 8\sqrt{2}$.