The Pythagorean Theorem
The most famous mathematical equation in history forms the bridge between pure geometry and distance calculation. It is the absolute backbone of trigonometry.
Introduction
In the previous lesson, we learned that the angles of a triangle are rigidly locked into a sum of . But what about the sides? If you lock in a Right Angle, the lengths of the three sides become mathematically tethered to each other by an inescapable rule.
Past Knowledge
You know how to evaluate basic exponents and roots (e.g., , ), and simplify basic algebraic equations.
Today's Goal
Use the Pythagorean Theorem to find the exact length of any missing side in a right triangle.
Future Success
This theorem will be transformed into the "Distance Formula" and later, the "Primary Pythagorean Identity" ().
Key Concepts
The Equation
The Pythagorean Theorem states that in any Right Triangle, the squares of the two shorter sides (legs) will exactly equal the square of the longest side (the hypotenuse).
- and (Legs): The two sides that meet to form the right angle. It doesn't matter which one is which.
- (Hypotenuse): The longest side of the triangle, which is always directly across from the right angle.
Worked Examples
Finding the Hypotenuse
Question: A ladder is leaning against a wall. The base of the ladder is placed exactly 5 feet away from the wall. The ladder touches the wall exactly 12 feet off the ground. How long is the ladder?
Step 1: Identify your components.
The wall and the ground perfectly meet to form a Right Angle. This means they are the "legs" ( and ). The ladder forms the diagonal "hypotenuse" ().
Step 2: Plug into the theorem.
Step 3: Square the numbers and add.
Step 4: Take the square root of both sides.
Final Answer: The ladder is 13 feet long.
Finding a Missing Leg
Question: The hypotenuse of a right triangle is 25 cm. One of the short legs is 7 cm. Find the length of the other leg.
Step 1: Assign your variables carefully!
The hypotenuse MUST be . The leg can be . We need to find .
Step 2: Square the numbers.
Step 3: Isolate the unknown.
Subtract 49 from both sides before taking any square roots!
Step 4: Take the square root.
Final Answer: The missing leg is 24 cm long.
Exact Radical Answers
Question: An isosceles right triangle has two equal legs measuring exactly inches. Find the exact length of the hypotenuse in simplified radical form (no decimals).
Step 1: Set up the equation.
Both legs are 6.
Step 2: Compute the sum.
Step 3: Simplify the radical.
We need . We don't want a long decimal like . We need to extract perfect squares.
Bring the perfect square out of the root as a .
Final Answer: The hypotenuse is exactly inches.
Common Pitfalls
The Non-Right Triangle Trap
It's tempting to use on every triangle you see because the formula is so memorable. But if the triangle doesn't have a perfect Right angle, the formula mathematically breaks and will give the wrong side length.
Later in the course, we will learn "The Law of Cosines," which is the upgraded version of this theorem that works on all triangles.
The "Distributive Root" Illusion
When solving for , students sometimes try to take the square root of safety individual pieces rather than the sum.
❌ (Never do this!)
Just try it with numbers: . But if you split it, you'd get , which is wrong! Always add the squares together first before rooting.
Real-Life Applications
GPS and Displacement (Crows vs. Taxis)
Imagine you need to walk 3 blocks East and 4 blocks North in a city. Because buildings block you, you must walk the "Taxicab Geometry" distance along the streets ( blocks).
However, a bird (or a plane, or a missile tracking system) can fly perfectly diagonal over the buildings. This "As the crow flies" displacement distance is the hypotenuse! The bird only has to fly blocks of absolute distance. GPS systems constantly run these right-triangle calculations in the background to tell you how far away a destination truly is.
Practice Quiz
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