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

Q: Why do both legs have the same expression x + 8?

In an isosceles right triangle, the two legs are always equal in length. The diagram shows both legs labeled as x + 8, confirming this is an isosceles triangle.

Q: How do I find the hypotenuse once I know the legs?

For any right triangle, use the Pythagorean theorem: $ c^2 = a^2 + b^2 $. But for isosceles right triangles, there's a shortcut: hypotenuse = leg × √2.

Q: What does the constraint x > -8 mean?

This constraint ensures the side length $ x + 8 $ is positive. Since lengths cannot be negative, we need $ x + 8 > 0 $, which means $ x > -8 $.

Q: Why do I get two solutions when solving (x + 8)² = 64?

Taking the square root gives $ x + 8 = ±8 $, so $ x = 0 $ or $ x = -16 $. But $ x = -16 $ violates the constraint $ x > -8 $, so we only use $ x = 0 $.

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find the lengths of the triangle's sides

00:03 We'll use the formula for calculating triangle area

00:07 (height times side) divided by 2

00:12 Sides are equal according to the given data

00:15 We'll substitute the area value according to the given data

00:25 We'll multiply by the denominator to eliminate the fraction

00:32 Taking the square root, remember we get 2 solutions

00:35 One positive and one negative

00:40 We'll isolate the unknown and find the solution for each possibility

00:46 This is one solution, but it doesn't fit the domain

00:53 This is the second solution, we'll substitute to find the sides

00:59 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

1

Understand the problem

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.

2

Step-by-step solution

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

3

Final Answer

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

Key Points to Remember

Essential concepts to master this topic

Isosceles Right Triangle: Both legs are equal, hypotenuse equals leg × √2
Area Formula: Area = ½ × leg² = ½ × (x + 8)² = 32
Verification: Check legs are 8 cm and area is ½ × 8² = 32 ✓

Common Mistakes

Avoid these frequent errors

Using wrong area formula for isosceles right triangle
Don't use Area = ½ × base × height with different values = wrong equation! Students often forget both legs are equal in isosceles triangles. Always use Area = ½ × leg² since both legs have the same length.

Practice Quiz

Test your knowledge with interactive questions

Find the value of the parameter x.

$$ 2x^2-7x+5=0 $$

FAQ

Everything you need to know about this question

Why do both legs have the same expression x + 8?

+

In an isosceles right triangle, the two legs are always equal in length. The diagram shows both legs labeled as x + 8, confirming this is an isosceles triangle.

How do I find the hypotenuse once I know the legs?

+

For any right triangle, use the Pythagorean theorem: $c^2 = a^2 + b^2$ . But for isosceles right triangles, there's a shortcut: hypotenuse = leg × √2.

What does the constraint x > -8 mean?

+

This constraint ensures the side length $x + 8$ is positive. Since lengths cannot be negative, we need $x + 8 > 0$ , which means $x > -8$ .

Why do I get two solutions when solving (x + 8)² = 64?

+

Taking the square root gives $x + 8 = ±8$ , so $x = 0$ or $x = -16$ . But $x = -16$ violates the constraint $x > -8$ , so we only use $x = 0$ .

How can I double-check my final answer?

+

Substitute back into the area formula: with legs of 8 cm each, $\text{Area} = \frac{1}{2} \times 8 \times 8 = 32$ ✓. The hypotenuse should be $8\sqrt{2}$ cm.

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