Pythagorean Theorem - Definition, Examples, Quiz, FAQ, Trivia
Learn about right triangles with easy explanations, visual examples, and practice activities
What is the Pythagorean Theorem?
The Pythagorean Theorem is a fundamental rule in geometry that helps us understand right triangles. A right triangle is any triangle that has one 90-degree angle (a right angle).
The theorem states that in a right triangle:
The area of the square on the hypotenuse (the side opposite the right angle) equals the sum of the areas of the squares on the other two sides.
This means if you know the lengths of any two sides of a right triangle, you can always find the length of the third side using this special relationship.
Key Concept
The hypotenuse is always the longest side of a right triangle, opposite the right angle.
The Pythagorean Formula
The Pythagorean Theorem can be written as a simple mathematical formula:
Pythagorean Formula
Where:
a and b are the lengths of the two legs (shorter sides)
c is the length of the hypotenuse (longest side)
a² means "a multiplied by a" (the area of the square on side a)
b² means "b multiplied by b" (the area of the square on side b)
c² means "c multiplied by c" (the area of the square on the hypotenuse)
The formula shows that when you add the areas of the squares on the two shorter sides, it equals the area of the square on the longest side.
Remember
The Pythagorean Theorem only works for right triangles. For other types of triangles, different rules apply.
Example Problems
Let's practice using the Pythagorean Theorem with some examples:
Example 1: A right triangle has legs of 3 cm and 4 cm. What is the length of the hypotenuse?
Solution:
a = 3, b = 4
a² + b² = c² → 3² + 4² = c² → 9 + 16 = c² → 25 = c² → c = √25 → c = 5 cm
Example 2: The hypotenuse of a right triangle is 13 inches, and one leg is 5 inches. Find the other leg.
Solution:
a = 5, c = 13
a² + b² = c² → 5² + b² = 13² → 25 + b² = 169 → b² = 169 - 25 → b² = 144 → b = √144 → b = 12 inches
Example 3: A ladder 10 feet long leans against a wall. The bottom is 6 feet from the wall. How high does the ladder reach?
Solution:
a = 6, c = 10
a² + b² = c² → 6² + b² = 10² → 36 + b² = 100 → b² = 100 - 36 → b² = 64 → b = √64 → b = 8 feet
Practice Tip
Always identify which side is the hypotenuse first (it's opposite the right angle and always the longest side).
Real-World Applications
The Pythagorean Theorem isn't just for math class - it's used in many real-world situations:
Construction: Builders use it to make sure corners are square (exactly 90 degrees) when framing houses.
Navigation: Pilots and sailors use it to calculate the shortest distance between two points.
Sports: The theorem helps determine the distance from home plate to second base in baseball (about 127.3 feet).
Computers: Used in computer graphics for calculating distances between points in 2D and 3D space.
Surveying: Land surveyors use it to measure distances that can't be measured directly.
Did You Know?
The Pythagorean Theorem is used in GPS technology to calculate your position on Earth!
Practice Quiz
Test your understanding with these practice questions. Choose the correct answer for each.
Frequently Asked Questions
Here are answers to common questions about the Pythagorean Theorem:
Math Trivia
Discover interesting facts about the Pythagorean Theorem:
Ancient Knowledge
The oldest known evidence of the Pythagorean relationship comes from a Babylonian clay tablet (Plimpton 322) dating to about 1800 BCE, over 1000 years before Pythagoras was born.
Many Proofs
There are over 400 known proofs of the Pythagorean Theorem, including one by U.S. President James Garfield. The simplest proof involves rearranging four identical right triangles within a square.
Space Mathematics
NASA uses the 3D version of the Pythagorean Theorem to calculate distances between spacecraft and planets, and to navigate space probes through the solar system.
Largest Demonstration
In 2017, 1,200 students in India formed a giant human right triangle to demonstrate the Pythagorean Theorem, setting a Guinness World Record for the largest such demonstration.
