Introduction to the Pythagorean Theorem
Understand the Pythagorean theorem (a² + b² = c²) through visual proofs using square areas. Apply it to find missing side lengths in right triangles and distances on the coordinate plane.
A lesson of Pre-Algebra, in Geometry Foundations, about 10 minutes.
By the end of this lesson, you will be able to find missing sides of right triangles, determine if three lengths form a right triangle, and apply the theorem to real-world problems.
The Theorem
For any right triangle with legs $a$ and $b$ and hypotenuse $c$:
$$a^2 + b^2 = c^2$$
The hypotenuse ($c$) is always the longest side, opposite the right angle.
Finding Missing Sides
Find the hypotenuse: $a = 5$, $b = 12$
$c^2 = 5^2 + 12^2 = 25 + 144 = 169 \to c = \sqrt{169} = 13$
Find a leg: $a = ?$, $b = 8$, $c = 17$
$a^2 + 64 = 289 \to a^2 = 225 \to a = 15$
Pythagorean triples (exact whole number answers):
$3\text{-}4\text{-}5$, $5\text{-}12\text{-}13$, $8\text{-}15\text{-}17$, $7\text{-}24\text{-}25$
Is It a Right Triangle?
Check if $a^2 + b^2 = c^2$ (use the longest side as $c$).
Example: 6, 8, 10 → $36 + 64 = 100$ ✓ Right triangle
Example: 4, 5, 7 → $16 + 25 = 41 \neq 49$ ✗ Not a right triangle
Real-World Application
A ladder 13 feet long leans against a wall. The base is 5 feet from the wall. How high does it reach?
$5^2 + h^2 = 13^2 \to 25 + h^2 = 169 \to h^2 = 144 \to h = 12$ feet
Summary
$a^2 + b^2 = c^2$ — hypotenuse is $c$ (longest side, opposite the right angle).
Finding $c$: add squares of legs, take square root.
Finding a leg: subtract the known leg squared from $c^2$, take square root.
Right triangle test: check if $a^2 + b^2 = c^2$.