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Pythagorean Theorem
Right Triangles

Learn to apply \( a^2 + b^2 = c^2 \) to find missing side lengths in right triangles: one of the most powerful tools in geometry.

Pre-Algebra

What Is a Right Triangle?

A
B
C
leg \(a\)
leg \(b\)
hypotenuse \(c\)
\(90^\circ\)

Parts of a Right Triangle

Legs
The two sides that form the right angle. Often labeled \( a \) and \( b \).
Hypotenuse
The side opposite the 90° angle. Always the longest side. Labeled \( c \).
Right Angle
Marked with a small square. Measures exactly \( 90^\circ \).

Why Does the Pythagorean Theorem Matter?

The Pythagorean Theorem is used everywhere: architecture, navigation, video games, and engineering. Anytime you need to find the distance between two points or a missing side of a right triangle, this theorem is your go-to tool.

Construction
Builders check that corners are square (90°) using 3-4-5 triangles.
Navigation
Pilots and sailors calculate straight-line distances using right triangles.

The Pythagorean Theorem

\(a\)
\(b\)
\(c\)
\( a^2 + b^2 = c^2 \)

Visual Proof: Squares on Sides

\( a^2 \)
\( b^2 \)
\( c^2 \)
\(a=4\)
\(b=3\)
\(c=5\)
\(16 + 9 = 25\)
\( 4^2 + 3^2 = 5^2 \)

Leg vs Hypotenuse: Key Distinction

Leg (\( a \) or \( b \)) Hypotenuse (\( c \))
Position Forms the right angle Opposite the right angle
Length Shorter Always the longest
Formula role Added: \( a^2 + b^2 \) Equals: \( = c^2 \)
Solving for it \( a = \sqrt{c^2 - b^2} \) \( c = \sqrt{a^2 + b^2} \)

How to Apply the Theorem

1
Confirm it's a right triangle. Look for the 90° angle or the square symbol.
2
Label the sides. Identify which side is the hypotenuse (\( c \)) and which are the legs (\( a \), \( b \)).
3
Plug into \( a^2 + b^2 = c^2 \). Substitute the known values.
4
Solve for the unknown. Use addition/subtraction, then take the square root.
5
Check your answer. Verify that \( c \) is the largest value, and substitute back to confirm.

Example 1: Finding the Hypotenuse

\(a = 6\)
\(b = 8\)
\(c = ?\)

Example 2: Finding a Leg

\(a = ?\)
\(b = 12\)
\(c = 13\)

Example 3: Non-Integer Answer

\(a = 5\)
\(b = 7\)
\(c = ?\)

Pythagorean Triples

A Pythagorean triple is a set of three whole numbers that satisfy \( a^2 + b^2 = c^2 \). Memorizing the most common triples saves time.

3, 4, 5
\( 9 + 16 = 25 \). Multiples: 6-8-10, 9-12-15, etc.
5, 12, 13
\( 25 + 144 = 169 \). Multiples: 10-24-26, etc.
8, 15, 17
\( 64 + 225 = 289 \). Less common but still useful.

Real-World Application

Ground: 12 ft
Wall: ? ft
Ladder: 15 ft
foot of ladder
top of wall

The Converse: Is It a Right Triangle?

The converse of the Pythagorean Theorem lets you test whether a triangle is a right triangle. If the sides satisfy \( a^2 + b^2 = c^2 \), then it must be a right triangle.

1
Order the sides from shortest to longest. The longest is your candidate hypotenuse \( c \).
2
Check: does \( a^2 + b^2 = c^2 \)?
3
If yes: it's a right triangle. If no: it's not a right triangle.

Common Mistakes to Avoid

✕ Adding All Three Sides
Students write \( a^2 + b^2 + c^2 \). Remember: \( c^2 \) is alone on one side of the equation.
✕ Forgetting the Square Root
Finding \( c^2 = 100 \) and answering 100 instead of \( \sqrt{100} = 10 \). Always take the final square root.
✕ Mixing Up Legs and Hypotenuse
Using \( a^2 + c^2 = b^2 \) with the hypotenuse as a leg. The longest side must be \( c \).

Summary: Quick Reference

Task Formula Example
Find hypotenuse \( c = \sqrt{a^2 + b^2} \) \( \sqrt{3^2 + 4^2} = 5 \)
Find a leg \( a = \sqrt{c^2 - b^2} \) \( \sqrt{13^2 - 12^2} = 5 \)
Check right triangle \( a^2 + b^2 \stackrel{?}{=} c^2 \) \( 5^2 + 12^2 = 169 = 13^2 \) ✓
Common triples 3-4-5, 5-12-13, 8-15-17, 7-24-25

Question 1 of 5

A right triangle has legs of length 9 and 12. What is the length of the hypotenuse?

Question 2 of 5

A right triangle has a hypotenuse of 10 and one leg of 6. What is the other leg?

Question 3 of 5

Which set of side lengths forms a right triangle?

Question 4 of 5

A 20-foot ladder leans against a wall with its base 16 feet from the wall. How high up the wall does the ladder reach?

Question 5 of 5

A right triangle has legs of length 1 and 1. What is the exact length of the hypotenuse?