Solve for X: Triangular Lengths in an Isosceles Right Triangle with Area 12.5 cm²

Q: Why are the two legs equal in an isosceles right triangle?

An isosceles right triangle has two equal sides (the legs) that meet at a 90° angle. The word 'isosceles' means equal sides, so both legs must have the same length.

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

Use the Pythagorean theorem: $ c^2 = a^2 + b^2 $. Since both legs equal 5, we get $ c^2 = 5^2 + 5^2 = 50 $, so $ c = \sqrt{50} = 5\sqrt{2} $.

Q: What if I get a negative value for x?

Check the constraint! The problem states $ x > -5 $, which means x must be greater than -5. Since we found x = 0, this satisfies the constraint perfectly.

Q: Why is the area formula ½ × leg² and not ½ × base × height?

Both formulas work! In an isosceles right triangle, the base and height are the same (both equal to the leg length). So ½ × base × height becomes ½ × leg × leg = ½ × leg².

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find the triangle side lengths

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

00:07 (height multiplied by side) divided by 2

00:15 Equal sides according to the given data

00:25 We'll substitute appropriate values according to the given data

00:38 We'll multiply by the denominator to eliminate the fraction

00:43 We'll extract the root, remember that when extracting a root we have 2 solutions

00:47 One positive and one negative

00:51 We'll isolate the unknown and find the solution for each possibility

00:54 This is one solution

00:58 This is the second solution, but doesn't fit the domain

01:01 We'll substitute the solution to find the sides

01:06 And this is the answer 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.

( $x>-5$ length measurements in cm).

Since the area of the triangle is 12.5.

Find the lengths of the sides of the triangle.

2

Step-by-step solution

The problem involves finding the side lengths of an isosceles right triangle given its area. Let's proceed with the solution.

Since the triangle is an isosceles right triangle, the two legs are equal, and the area $A$ is provided by the formula: $A = \frac{1}{2} \cdot \text{base} \cdot \text{height}$ For this triangle, both the base and the height are $x + 5$ .

The area is given as 12.5, so we set up the equation: $12.5 = \frac{1}{2} \cdot (x + 5) \cdot (x + 5)$ $12.5 = \frac{1}{2} \cdot (x + 5)^2$

Multiply both sides by 2 to solve for $(x + 5)^2$ : $25 = (x + 5)^2$

Take the square root of both sides: $x + 5 = \sqrt{25}$ $x + 5 = 5$

Solve for $x$ : $x = 5 - 5$ $x = 0$

Therefore, the length of each leg of the triangle is $x + 5 = 5$ cm.

For the hypotenuse $c$ , use the Pythagorean theorem $c^2 = a^2 + a^2$ , where $a = x + 5 = 5$ : $c^2 = 5^2 + 5^2 = 25 + 25 = 50$ $c = \sqrt{50} = 5\sqrt{2}$

Thus, the lengths of the sides of the triangle are $5$ , $5$ , and $5\sqrt{2}$ .

Therefore, the correct solution is $5, 5, 5\sqrt{2}$ .

3

Final Answer

$5,5,5\sqrt{2}$

Key Points to Remember

Essential concepts to master this topic

Triangle Property: Isosceles right triangle has two equal legs and 90° angle
Technique: Use Area = ½ × leg² formula: 12.5 = ½(x+5)²
Check: Verify legs are 5 cm and hypotenuse is 5√2 using Pythagorean theorem ✓

Common Mistakes

Avoid these frequent errors

Using wrong area formula for isosceles right triangle
Don't use Area = ½ × base × height with different values for base and height = wrong equation setup! In isosceles right triangles, both legs are equal, so the formula becomes Area = ½ × leg². Always recognize that both legs have the same length (x+5).

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 are the two legs equal in an isosceles right triangle?

+

An isosceles right triangle has two equal sides (the legs) that meet at a 90° angle. The word 'isosceles' means equal sides, so both legs must have the same length.

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

+

Use the Pythagorean theorem: $c^2 = a^2 + b^2$ . Since both legs equal 5, we get $c^2 = 5^2 + 5^2 = 50$ , so $c = \sqrt{50} = 5\sqrt{2}$ .

What if I get a negative value for x?

+

Check the constraint! The problem states $x > -5$ , which means x must be greater than -5. Since we found x = 0, this satisfies the constraint perfectly.

Can I solve this without using the area formula?

+

No - the area is the key information given! Without using Area = 12.5, you can't determine the specific value of x. The area constraint is what makes this problem solvable.

Why is the area formula ½ × leg² and not ½ × base × height?

+

Both formulas work! In an isosceles right triangle, the base and height are the same (both equal to the leg length). So ½ × base × height becomes ½ × leg × leg = ½ × leg².

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Solve for X: Triangular Lengths in an Isosceles Right Triangle with Area 12.5 cm²

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