The Pythagorean theorem applies to every right triangle. It connects the three sides through one equation. Once you understand it, you can find missing sides, classify triangles, calculate diagonals, and solve coordinate distances using the same rule.
The Pythagorean Theorem Explained
When Do You Add and When Do You Subtract?
Decide which case applies before writing any equation.
| Situation | Table Header | Table Header |
|---|---|---|
|
Both legs known, find hypotenuse |
c = √(a² + b²) |
Add the squares |
|
Hypotenuse and one leg known, find a leg |
a = √(c² − b²) |
Subtract the squares |
|
Three sides given, classify the triangle |
Compare a² + b² with c² |
Add, then compare |
What Does the Pythagorean Theorem Actually Mean?
The theorem is a statement about area. If you draw a square on each side of a right triangle, the combined area of the two smaller squares equals the area of the square on the hypotenuse.
3² + 4² = 5²
9 + 16 = 25
LHS = RHS
The two legs, a and b, meet at the right angle. The hypotenuse c is opposite the right angle and is always the longest side. Every formula in this topic begins by identifying c correctly.
How Do You Find the Hypotenuse?
To find the hypotenuse, square both legs (a , b), add them, and take the square root. The hypotenuse (c) is always the longest side.
Example 1. Whole-number answer
- Legs – 3 and 4
- c² = 3² + 4² = 9 + 16 = 25
- c = √25 = 5
Example 2. Irrational answer
- Legs – 5 and 7
- c² = 5² + 7² = 25 + 49 = 74
- c = √74
Since 74 is not a perfect square, leave the answer as √74 unless the problem asks for a decimal.
Example 3. Scaled triple
- Legs – 9 and 12
- c² = 9² + 12² = 81 + 144 = 225
- c = √225 = 15
9 and 12 both divide by 3 to give 3 and 4, which is the 3-4-5 triple. The hypotenuse is 5 × 3 = 15. No full calculation needed.
Question
A ladder leans against a wall and reaches the upper right corner of a window 12 feet above the ground. The window is 9 feet wide and the base of the ladder sits directly below the window’s left edge. How long is the ladder?
Answer choices
A. 12 B. 13 C. 15 D. 21
Solution
- Vertical leg = 12 ft, Horizontal leg = 9 ft, Ladder = hypotenuse
- L² = 9² + 12² = 81 + 144 = 225
- L = √225 = 15 ft
Triple shortcut
9 ÷ 3 = 3 and 12 ÷ 3 = 4, matching the 3-4-5 triple scaled by 3. The hypotenuse is 5 × 3 = 15 instantly.
Why the other choices are wrong
A (12) mistakes a leg for the hypotenuse, B (13) incorrectly applies the 5-12-13 triple and D (21) adds the legs directly without squaring.
Think10x.ai Video Explaining the Ladder Problem
How Do You Find a Missing Leg?
Subtract the square of the known leg from the square of the hypotenuse, then take the square root.
- Hypotenuse – 13
- Known leg – 5
- a² = 13² − 5² = 169 − 25 = 144
- a = √144 = 12
If the result comes out longer than the hypotenuse, the setup is wrong. Check which side is actually the hypotenuse before substituting.
Missing Leg Worked Example and Solution
Question
A right triangle has a hypotenuse of 13√2 and one leg of 11. Find the length of the other leg.
Answer choices
A. √217 B. √219 C. 15 D. 13
Solution
- Write the missing-leg formula (a² = c² − b²)
- Square the radical hypotenuse – (13√2)² = 13² × (√2)² = 169 × 2 = 338
- Square the known leg – 11² = 121
- Subtract and solve – a² = 338 − 121 = 217
- a = √217
Key rule
When the hypotenuse is written as n√k, square both parts separately – (n√k)² = n² · k. Squaring only the coefficient 13 and ignoring √2 is the most common error on this type of question.
Why the other choices are wrong
B (√219) comes from a slip when squaring 11, C (15) forces a familiar triple and ignores the radical and D (13) squares only the coefficient.
Think10x.ai Video Explaining Missing Leg Problem
What Are Pythagorean Triples?
A Pythagorean triple is a set of three positive integers that satisfy a² + b² = c². Recognizing one in a problem means you can state the answer immediately, without any calculation.
| a | b | c | Notes |
|---|---|---|---|
|
3 |
4 |
5 |
Most common |
|
5 |
12 |
13 |
Common on tests |
|
8 |
15 |
17 |
Harder word problems |
|
7 |
24 |
25 |
Useful extra triple |
|
6 |
8 |
10 |
3 – 4 – 5 × 2 |
|
9 |
12 |
15 |
3 – 4 – 5 × 3 |
|
10 |
24 |
26 |
5 – 12 – 13 × 2 |
|
15 |
20 |
25 |
3 – 4 – 5 × 5 |
Scale-factor shortcut
If both legs divide by the same number to match a known triple, multiply the third side by that same factor.
How Do You Use the Converse to Classify a Triangle?
The converse helps you decide whether a triangle is right, acute, or obtuse. Always identify the longest side first and label it as c before applying the test.
| If | Then the triangle is |
|---|---|
|
a² + b² = c² |
Right |
|
a² + b² > c² |
Acute |
|
a² + b² < c² |
Obtuse |
Worked example
- Sides – 7, 24, 25.
- Since 25 is the longest side, let c = 25.
- 7² + 24² = 49 + 576 = 625
- 25² = 625
Because both sides are equal, this is a right triangle.
How Do You Find the Diagonal of a Rectangle?
A diagonal divides a rectangle into two right triangles. The two side lengths are the legs and the diagonal is the hypotenuse.
d = √(l² + w²)
Example 1. Room Diagonal
Length is 15 ft and width is 20 ft
- d² = 15² + 20² = 225 + 400 = 625
- d = √625 = 25 ft
15 and 20 are the 3-4-5 triple multiplied by 5, so the diagonal is 5 × 5 = 25 immediately.
Example 2. Screen Diagonal
Width is 48 in and height is 27 in
- d² = 48² + 27² = 2,304 + 729 = 3,033
- d = √3,033 = 55.1 in
Screen sizes (TVs, monitors, tablets, phones) are always measured diagonally using this formula.
Where Is the Pythagorean Theorem Used in Real Life?
1. Ladders and ramps
The foot-to-wall distance and the height reached are the two legs; the ladder is the hypotenuse.
Ladder reaches 16 ft up a wall
Base is 12 ft from wall
- L² = 12² + 16² = 144 + 256 = 400
- L = √400 = 20 ft
2. Construction (Checking Right Angles)
Builders use the 3-4-5 rule to verify a 90° corner. Mark 3 ft along one wall and 4 ft along the adjacent wall. If the diagonal is 5 ft, the corner is a perfect right angle. The same check works at any scale: 6-8-10, 9-12-15, and so on.
3. Distance on a coordinate grid
The horizontal and vertical differences between two points are the legs. The straight-line distance is the hypotenuse.
- d = √((x₂ − x₁)² + (y₂ − y₁)²)
- Points (1, 2) and (7, 10)
- Horizontal – 7 − 1 = 6
- Vertical – 10 − 2 = 8
- d = √(36 + 64)
- √100 = 10
Common Traps and How to Avoid Them
| Trap | Fix |
|---|---|
|
Adding legs before squaring |
Square first, then add |
|
Forgetting the square root |
Always take the square root at the end |
|
Subtracting to find the hypotenuse |
Only subtract when finding a missing leg |
|
Mislabeling the hypotenuse |
It is opposite the right angle, always the longest side, and the side you assign as c in all formulas |
|
Squaring a radical incorrectly |
Square both parts separately: (n√k)² = n²k |
|
Rounding too early |
Round only at the final step |
Frequently Asked Questions
It states that in any right triangle, a² + b² = c², where a and b are the legs and c is the hypotenuse. It only applies to right triangles.
Use a = √(c² − b²). Square the hypotenuse, subtract the square of the known leg, then take the square root. Confirm which side is the hypotenuse before substituting.
Start with 3-4-5 and 5-12-13. Then add 8-15-17 and 7-24-25. Any multiple of a triple is also a triple: 6-8-10, 9-12-15, 10-24-26, and so on.
Use d = √(l² + w²). Square the length and width, add them, then take the square root. This is the same structure as finding the hypotenuse of a right triangle.
Square both parts separately. (13√2)² = 13² × (√2)² = 169 × 2 = 338. Squaring only the coefficient and ignoring the radical is the most common error.
The distance formula d = √((x₂ − x₁)² + (y₂ − y₁)²) is the Pythagorean theorem applied to coordinate geometry. The horizontal and vertical differences are the legs; the straight-line distance is the hypotenuse.
It only applies to right triangles. For triangles without a right angle, use the Law of Cosines or Law of Sines instead.
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