The Pythagorean Theorem
Grade 8 · Math · Study Guide
Discover the relationship between the sides of a right triangle and use it to solve real-world problems.
The Pythagorean Theorem states that in any right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides: a² + b² = c². The hypotenuse (c) is always the longest side and always opposite the right angle. The other two sides (a and b) are called legs.
To find a missing side, set up the equation and solve. If you're looking for the hypotenuse: c = √(a² + b²). If you're looking for a leg: a = √(c² - b²). Always check that your triangle actually has a right angle before applying this theorem.
The theorem also lets you check whether a triangle is a right triangle — if a² + b² = c², it is! Common Pythagorean triples (integer solutions) like 3-4-5, 5-12-13, and 8-15-17 are worth memorizing. The Pythagorean Theorem has applications everywhere: construction, navigation, computer graphics, and even your TV screen size.
| Leg a | Leg b | Hypotenuse c | Check: a²+b²=c² |
|---|---|---|---|
| 3 | 4 | 5 | 9+16=25 ✓ |
| 5 | 12 | 13 | 25+144=169 ✓ |
| 8 | 15 | 17 | 64+225=289 ✓ |
| 7 | 24 | 25 | 49+576=625 ✓ |
Key terms
- Hypotenuse
- The longest side of a right triangle; opposite the right angle; the 'c' in a²+b²=c².
- Legs
- The two shorter sides of a right triangle that form the right angle.
- Pythagorean Triple
- A set of three positive integers that satisfy a²+b²=c², e.g., 3-4-5.
- Right Angle
- A 90° angle; marked with a small square in diagrams.
- Radical (√)
- The square root symbol; √25 = 5 because 5² = 25.
Flashcards
- Pythagorean Theorem
- a² + b² = c²; applies to right triangles where c is the hypotenuse.
- Finding the hypotenuse
- c = √(a² + b²); add the squares of both legs, then take the square root.
- Finding a missing leg
- a = √(c² − b²); subtract the known leg squared from hypotenuse squared.
- Pythagorean triple
- Three integers that satisfy a²+b²=c². Classic example: 3, 4, 5.
- Checking for a right triangle
- If a²+b²=c² is true for the three sides, the triangle has a right angle.
- Hypotenuse location
- Always the side opposite the right angle — and always the longest side.
Practice questions
A right triangle has legs of 6 and 8. What is the hypotenuse?
- 10
- 12
- 14
- √100
Show answer
Answer: 10
6²+8²=36+64=100; √100=10. This is the 3-4-5 triple scaled by 2.
In the Pythagorean Theorem, 'c' always represents a leg of the triangle.
Show answer
Answer: False
c is the hypotenuse — the longest side opposite the right angle. a and b are the legs.
The Pythagorean Theorem formula is a² + b² = ___.
Show answer
Answer: c²
The theorem states the sum of the squares of the legs equals the square of the hypotenuse.
A ladder 10 ft long leans against a wall. Its base is 6 ft from the wall. How high does it reach?
- 6 ft
- 8 ft
- √64 ft
- Both B and C
Show answer
Answer: Both B and C
6²+h²=10² → h²=100-36=64 → h=√64=8 ft. Both 8 ft and √64 ft are equivalent correct answers.
Is a triangle with sides 5, 7, and 9 a right triangle?
- Yes, because 5+7=12>9
- No, because 5²+7²≠9²
- Yes, because 5²+9²=7²
- Cannot be determined
Show answer
Answer: No, because 5²+7²≠9²
5²+7²=25+49=74, but 9²=81. Since 74≠81, this is not a right triangle.
Fun facts
- Ancient Egyptians used the 3-4-5 rope trick to create perfect right angles when building the pyramids — no protractors needed!
- Over 370 different proofs of the Pythagorean Theorem exist, including one by U.S. President James Garfield in 1876.
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