Pythagorean Theorem

Relationships in right triangles

Written and reviewed by the Study Mondo Education TeamLast updated February 16, 2026

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Pythagorean Theorem

The Theorem

In a right triangle, the square of the hypotenuse equals the sum of the squares of the legs.

$a^2 + b^2 = c^2$

Where:

$a$ and $b$ are the legs
$c$ is the hypotenuse (longest side, opposite the right angle)

Using the Theorem

To find the hypotenuse: $c = \sqrt{a^2 + b^2}$

To find a leg: $a = \sqrt{c^2 - b^2}$

Pythagorean Triples

Sets of three positive integers that satisfy $a^2 + b^2 = c^2$ :

Common triples:

$3, 4, 5$
$5, 12, 13$
$8, 15, 17$
$7, 24, 25$

Any multiple works too: $6, 8, 10$ (multiply $3, 4, 5$ by 2)

Converse

If $a^2 + b^2 = c^2$ , then the triangle is a right triangle.

Distance Formula

The Pythagorean Theorem leads to the distance formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

📚 Practice Problems

1Problem 1easy

❓ Question:

Find the length of the hypotenuse if the legs of a right triangle are 3 and 4.

💡 Show Solution

Use $a^2 + b^2 = c^2$ :

$3^2 + 4^2 = c^2$

$9 + 16 = c^2$

$25 = c^2$

$c = 5$

Answer: The hypotenuse is $5$

2Problem 2easy

❓ Question:

Find the length of the hypotenuse if the legs of a right triangle are 3 and 4.

💡 Show Solution

Use $a^2 + b^2 = c^2$ :

$3^2 + 4^2 = c^2$

$9 + 16 = c^2$

$25 = c^2$

$c = 5$

Answer: The hypotenuse is $5$

3Problem 3medium

❓ Question:

A right triangle has a hypotenuse of 13 and one leg of 5. Find the other leg.

💡 Show Solution

Use $a^2 + b^2 = c^2$ :

$5^2 + b^2 = 13^2$

$25 + b^2 = 169$

$b^2 = 144$

$b = 12$

Answer: The other leg is $12$

4Problem 4medium

❓ Question:

A right triangle has a hypotenuse of 13 and one leg of 5. Find the other leg.

💡 Show Solution

Use $a^2 + b^2 = c^2$ :

$5^2 + b^2 = 13^2$

$25 + b^2 = 169$

$b^2 = 144$

$b = 12$

Answer: The other leg is $12$

5Problem 5hard

❓ Question:

Is a triangle with sides 7, 24, and 25 a right triangle?

💡 Show Solution

Use the converse of the Pythagorean Theorem.

Check if $a^2 + b^2 = c^2$ (where $c = 25$ is the longest side):

$7^2 + 24^2 = 49 + 576 = 625$

$25^2 = 625$

Since $7^2 + 24^2 = 25^2$ , the triangle is a right triangle.

Answer: Yes, it is a right triangle (and $7, 24, 25$ is a Pythagorean triple)

6Problem 6hard

❓ Question:

Is a triangle with sides 7, 24, and 25 a right triangle?

💡 Show Solution

Use the converse of the Pythagorean Theorem.

Check if $a^2 + b^2 = c^2$ (where $c = 25$ is the longest side):

$7^2 + 24^2 = 49 + 576 = 625$

$25^2 = 625$

Since $7^2 + 24^2 = 25^2$ , the triangle is a right triangle.

Answer: Yes, it is a right triangle (and $7, 24, 25$ is a Pythagorean triple)

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❓ Frequently Asked Questions

What is Pythagorean Theorem?▾

Relationships in right triangles

How can I study Pythagorean Theorem effectively?▾

Start by reading the study notes and working through the examples on this page. Then use the flashcards to test your recall. Practice with the 6 problems provided, checking solutions as you go. Regular review and active practice are key to retention.

Is this Pythagorean Theorem study guide free?▾

Yes — all study notes, flashcards, and practice problems for Pythagorean Theorem on Study Mondo are free to access. No account is needed.

What course covers Pythagorean Theorem?▾

Pythagorean Theorem is part of the Geometry course on Study Mondo, specifically in the Triangles section. You can explore the full course for more related topics and practice resources.

Are there practice problems for Pythagorean Theorem?▾

Yes, this page includes 6 practice problems with detailed solutions. Each problem includes a step-by-step explanation to help you understand the approach.

Pythagorean Theorem

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Pythagorean Theorem

The Theorem

Using the Theorem

Pythagorean Triples

Converse

Distance Formula

📚 Practice Problems

1Problem 1easy

❓ Question:

2Problem 2easy

❓ Question:

3Problem 3medium

❓ Question:

4Problem 4medium

❓ Question:

5Problem 5hard

❓ Question:

6Problem 6hard

❓ Question:

Practice Lab

Practice with Flashcards

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📌 Related Topics in Triangles

❓ Frequently Asked Questions