Pythagorean Theorem and Geometry
Grade 9 · Math · Study Guide
Apply the Pythagorean Theorem to find missing side lengths, identify Pythagorean triples, and use geometry in 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 two legs: a² + b² = c², where c is always the hypotenuse (the side opposite the right angle). For example, if a = 3 and b = 4, then c² = 9 + 16 = 25, so c = 5.
You can also use the theorem to find a leg: if one leg and the hypotenuse are known, rearrange to a² = c² − b². Pythagorean triples are whole-number sets that satisfy the theorem, such as (3, 4, 5), (5, 12, 13), and (8, 15, 17). Any multiple of a triple also works — (6, 8, 10) is just (3, 4, 5) doubled.
The converse of the Pythagorean Theorem is equally useful: if a² + b² = c², the triangle is a right triangle. If a² + b² > c², it is acute; if a² + b² < c², it is obtuse. The distance formula — used to find distance between two coordinate points — is derived directly from the Pythagorean Theorem.
| Leg a | Leg b | Hypotenuse c | Verification |
|---|---|---|---|
| 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
- Pythagorean Theorem
- a² + b² = c²; applies to right triangles where c is the hypotenuse
- Hypotenuse
- The longest side of a right triangle, opposite the right angle
- Leg
- Either of the two shorter sides of a right triangle that form the right angle
- Pythagorean triple
- A set of three whole numbers that satisfies a² + b² = c², e.g., (3, 4, 5)
- Converse of Pythagorean Theorem
- If a² + b² = c², the triangle is a right triangle
- Distance formula
- d = √((x₂−x₁)² + (y₂−y₁)²) — derived from the Pythagorean Theorem
Flashcards
- Pythagorean Theorem
- a² + b² = c² for any right triangle
- Hypotenuse
- The longest side; opposite the right angle
- Pythagorean triple
- Three whole numbers satisfying a² + b² = c²; e.g., 3-4-5
- Distance formula
- d = √((x₂−x₁)² + (y₂−y₁)²)
- Converse of Pythagorean Theorem
- If a² + b² = c², the triangle is a right triangle
- Leg of a right triangle
- One of the two shorter sides that meet at the right angle
Practice questions
A right triangle has legs of 6 and 8. What is the length of the hypotenuse?
- 10
- 12
- 14
- 7
Show answer
Answer: 10
c² = 6² + 8² = 36 + 64 = 100; c = √100 = 10.
In a right triangle with hypotenuse 13 and one leg 5, the other leg is 12.
Show answer
Answer: True
b² = 13² − 5² = 169 − 25 = 144; b = √144 = 12. ✓
The Pythagorean Theorem formula is a² + b² = ___.
Show answer
Answer: c²
The full statement is a² + b² = c², where c is the hypotenuse.
What is the distance between points (0, 0) and (3, 4)?
- 4
- 5
- 7
- 3
Show answer
Answer: 5
d = √(3² + 4²) = √(9 + 16) = √25 = 5.
A right triangle has a hypotenuse of 17 and one leg of 8. What is the other leg?
- 13
- 15
- 12
- 9
Show answer
Answer: 15
b² = 17² − 8² = 289 − 64 = 225; b = √225 = 15.
Fun facts
- The Pythagorean Theorem was known to Babylonian mathematicians over 1,000 years before Pythagoras was born!
- Builders today still use the 3-4-5 triangle to make perfectly square corners when laying foundations.
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