1. The Pythagorean Theorem

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

The Formula:

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

Key Terms:

• Legs (a and b): The two sides that form the right angle
• Hypotenuse (c): The longest side, opposite the right angle
• Right triangle: A triangle with one 90° angle

Important Notes:

• The Pythagorean theorem ONLY works for right triangles
• The hypotenuse (c) is always the longest side
• The hypotenuse is always opposite the right angle
• The legs are the two shorter sides that form the right angle

Visual Representation:

In a right triangle with legs \( a \) and \( b \), and hypotenuse \( c \):

• The area of the square on leg \( a \) = \( a^2 \)
• The area of the square on leg \( b \) = \( b^2 \)
• The area of the square on hypotenuse \( c \) = \( c^2 \)
• Therefore: \( a^2 + b^2 = c^2 \)

2. Find the Length of the Hypotenuse

Formula:

\( c = \sqrt{a^2 + b^2} \)

where \( a \) and \( b \) are the legs, and \( c \) is the hypotenuse

Steps to Find the Hypotenuse:

1. Square both leg lengths
2. Add the squared values
3. Take the square root of the sum

Examples:

Example 1: Find the hypotenuse if the legs are 3 and 4.

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

\( c^2 = 3^2 + 4^2 \)

\( c^2 = 9 + 16 = 25 \)

\( c = \sqrt{25} = 5 \)

Example 2: Find the hypotenuse if the legs are 5 and 12.

\( c^2 = 5^2 + 12^2 = 25 + 144 = 169 \)

\( c = \sqrt{169} = 13 \)

Example 3: Find the hypotenuse if the legs are 6 and 8.

\( c^2 = 6^2 + 8^2 = 36 + 64 = 100 \)

\( c = \sqrt{100} = 10 \)

Example 4: Find the hypotenuse if the legs are 7 and 9.

\( c^2 = 7^2 + 9^2 = 49 + 81 = 130 \)

\( c = \sqrt{130} \approx 11.4 \)

3. Find the Missing Leg Length

Formulas:

\( a = \sqrt{c^2 - b^2} \)

or

\( b = \sqrt{c^2 - a^2} \)

Steps to Find a Missing Leg:

1. Square the hypotenuse
2. Square the known leg
3. Subtract the squared leg from the squared hypotenuse
4. Take the square root of the result

Examples:

Example 1: Find the missing leg if the hypotenuse is 10 and one leg is 6.

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

\( a^2 + 6^2 = 10^2 \)

\( a^2 + 36 = 100 \)

\( a^2 = 100 - 36 = 64 \)

\( a = \sqrt{64} = 8 \)

Example 2: Find the missing leg if the hypotenuse is 13 and one leg is 5.

\( b^2 = c^2 - a^2 = 13^2 - 5^2 = 169 - 25 = 144 \)

\( b = \sqrt{144} = 12 \)

Example 3: Find the missing leg if the hypotenuse is 15 and one leg is 9.

\( a^2 = 15^2 - 9^2 = 225 - 81 = 144 \)

\( a = \sqrt{144} = 12 \)

Example 4: Find the missing leg if the hypotenuse is 20 and one leg is 16.

\( b^2 = 20^2 - 16^2 = 400 - 256 = 144 \)

\( b = \sqrt{144} = 12 \)

4. Find the Perimeter Using Pythagorean Theorem

Perimeter Formula:

\( P = a + b + c \)

Perimeter = sum of all three sides

Steps to Find Perimeter:

1. Use the Pythagorean theorem to find the missing side
2. Add all three sides together

Examples:

Example 1: Find the perimeter of a right triangle with legs 3 and 4.

Step 1: Find hypotenuse: \( c = \sqrt{3^2 + 4^2} = \sqrt{25} = 5 \)

Step 2: Add all sides: \( P = 3 + 4 + 5 = 12 \)

Example 2: Find the perimeter of a right triangle with legs 5 and 12.

Find hypotenuse: \( c = \sqrt{5^2 + 12^2} = \sqrt{169} = 13 \)

Perimeter: \( P = 5 + 12 + 13 = 30 \)

Example 3: Find the perimeter of a right triangle with hypotenuse 10 and one leg 6.

Find other leg: \( b = \sqrt{10^2 - 6^2} = \sqrt{64} = 8 \)

Perimeter: \( P = 6 + 8 + 10 = 24 \)

Example 4: Find the perimeter of a right triangle with legs 8 and 15.

Find hypotenuse: \( c = \sqrt{8^2 + 15^2} = \sqrt{289} = 17 \)

Perimeter: \( P = 8 + 15 + 17 = 40 \)

5. Pythagorean Theorem: Word Problems

Common Types of Word Problems:

• Ladder problems (ladder leaning against a wall)
• Distance problems (diagonal distance across a field)
• Screen size problems (TV or phone diagonal)
• Navigation problems (finding shortest distance)
• Sports problems (baseball diamond, soccer field)

Steps to Solve Word Problems:

1. Read the problem carefully and identify the right triangle
2. Label the known sides (legs or hypotenuse)
3. Identify what you need to find
4. Use the appropriate Pythagorean theorem formula
5. Solve and include units in your answer

Examples:

Example 1 (Ladder Problem): A 25-foot ladder leans against a wall. The base of the ladder is 7 feet from the wall. How high up the wall does the ladder reach?

Identify: Ladder = hypotenuse (25 ft), base = one leg (7 ft), height = other leg (unknown)

\( h^2 = 25^2 - 7^2 = 625 - 49 = 576 \)

\( h = \sqrt{576} = 24 \) feet

Example 2 (Distance Problem): A person walks 12 meters east, then 5 meters north. How far are they from their starting point?

Identify: Forms a right triangle with legs 12 m and 5 m

\( c = \sqrt{12^2 + 5^2} = \sqrt{144 + 25} = \sqrt{169} = 13 \) meters

Example 3 (Screen Size): A rectangular TV screen is 24 inches wide and 18 inches tall. What is the diagonal measurement (screen size)?

\( d = \sqrt{24^2 + 18^2} = \sqrt{576 + 324} = \sqrt{900} = 30 \) inches

Example 4 (Baseball Diamond): A baseball diamond is a square with sides of 90 feet. What is the distance from home plate to second base?

Diagonal of square: \( d = \sqrt{90^2 + 90^2} = \sqrt{16200} \approx 127.3 \) feet

Example 5 (Ramp Problem): A wheelchair ramp is 20 feet long and rises 2 feet. How far is the base of the ramp from the building?

\( b = \sqrt{20^2 - 2^2} = \sqrt{400 - 4} = \sqrt{396} \approx 19.9 \) feet

6. Pythagorean Triples

Definition: A Pythagorean triple is a set of three positive integers \( a \), \( b \), and \( c \) that satisfy the equation \( a^2 + b^2 = c^2 \).

Common Pythagorean Triples:

Multiples of Pythagorean Triples:

If \( (a, b, c) \) is a Pythagorean triple, then \( (ka, kb, kc) \) is also a Pythagorean triple for any positive integer \( k \).

Examples:

• From (3, 4, 5): multiply by 2 → (6, 8, 10)
• From (3, 4, 5): multiply by 3 → (9, 12, 15)
• From (5, 12, 13): multiply by 2 → (10, 24, 26)

Why Pythagorean Triples Are Useful:

• Provide quick answers without calculation
• Help identify right triangles immediately
• Used to check if answers are reasonable
• Save time on tests and homework

7. Converse of the Pythagorean Theorem

Statement: If \( a^2 + b^2 = c^2 \) for the sides of a triangle, then the triangle IS a right triangle.

Purpose:

The converse helps us determine whether a triangle with given side lengths is a right triangle.

Steps to Test if a Triangle is a Right Triangle:

1. Identify the longest side (this would be the hypotenuse if it's a right triangle)
2. Square all three sides
3. Check if: (leg₁)² + (leg₂)² = (longest side)²
4. If yes, it IS a right triangle. If no, it is NOT a right triangle

Three Cases:

Examples:

Example 1: Is a triangle with sides 9, 12, and 15 a right triangle?

Longest side = 15

Check: \( 9^2 + 12^2 = 15^2 \) ?

\( 81 + 144 = 225 \)

\( 225 = 225 \) ✓

Yes, it is a right triangle.

Example 2: Is a triangle with sides 5, 6, and 7 a right triangle?

Longest side = 7

Check: \( 5^2 + 6^2 = 7^2 \) ?

\( 25 + 36 = 49 \)

\( 61 \neq 49 \) ✗

No, it is NOT a right triangle.

Example 3: Is a triangle with sides 8, 15, and 17 a right triangle?

Check: \( 8^2 + 15^2 = 17^2 \) ?

\( 64 + 225 = 289 \)

\( 289 = 289 \) ✓

Yes, it is a right triangle. (This is a Pythagorean triple!)

Example 4: Is a triangle with sides 7, 10, and 12 a right triangle?

Check: \( 7^2 + 10^2 = 12^2 \) ?

\( 49 + 100 = 144 \)

\( 149 \neq 144 \) ✗

Since \( 149 > 144 \), this is an acute triangle.

8. Pythagorean Theorem and Distance Formula

Connection: The distance formula is derived from the Pythagorean theorem!

Distance Formula:

\( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \)

How It Works:

• Horizontal distance = \( |x_2 - x_1| \) (one leg)
• Vertical distance = \( |y_2 - y_1| \) (other leg)
• Diagonal distance = hypotenuse
• Apply Pythagorean theorem to find the diagonal

Example:

Find the distance between points \( A(1, 2) \) and \( B(4, 6) \).

Horizontal distance: \( |4 - 1| = 3 \)

Vertical distance: \( |6 - 2| = 4 \)

Distance: \( d = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \)

Quick Reference: Pythagorean Theorem Formulas

Common Pythagorean Triples to Memorize

(3, 4, 5) • (5, 12, 13) • (8, 15, 17) • (7, 24, 25)

💡 Key Tips for Pythagorean Theorem

• ✓ Only works for right triangles! Must have a 90° angle
• ✓ Hypotenuse (c) is ALWAYS the longest side opposite the right angle
• ✓ Legs (a and b) are the two sides forming the right angle
• ✓ Formula: \( a^2 + b^2 = c^2 \) — memorize this!
• ✓ When finding a leg: Subtract, then take square root
• ✓ When finding hypotenuse: Add, then take square root
• ✓ Memorize common Pythagorean triples: (3,4,5), (5,12,13), (8,15,17)
• ✓ Multiples also work: (6,8,10), (9,12,15), (15,20,25)
• ✓ Converse tests for right triangles: If \( a^2 + b^2 = c^2 \), it's a right triangle
• ✓ Always identify the longest side first when using converse
• ✓ In word problems, draw a diagram! It helps visualize the triangle
• ✓ Don't forget units in your final answer