Investigation 2: Pythagorean Theorem

New Concept

This theorem defines a fundamental relationship in all right triangles, connecting the lengths of the three sides with a simple, powerful equation.

If a triangle is a right triangle, then the sum of the squares of the legs equals the square of the hypotenuse.

$$a^2 + b^2 = c^2$$

What’s next

Property

If a triangle is a right triangle, then the sum of the squares of the legs equals the square of the hypotenuse.

Examples

Find the hypotenuse $c$ when the legs are $a=9$ and $b=12$: $9^2 + 12^2 = 81 + 144 = 225$, so $c = \sqrt{225} = 15$.
Find the leg $a$ when the other leg is $b=24$ and the hypotenuse is $c=26$: $a^2 + 24^2 = 26^2$, so $a^2 = 676 - 576 = 100$, and $a=10$.
For a square with a side length of 1, find the diagonal $d$: $1^2 + 1^2 = d^2$, so $d^2 = 2$ and the diagonal is $d = \sqrt{2}$.

Explanation

Think of it as a magic puzzle! If you draw squares on the two shorter sides (legs) of a right triangle, their combined area perfectly equals the area of the big square on the longest side (the hypotenuse). This ancient rule, named after the Greek mathematician Pythagoras, helps you find any missing side length on a right triangle.

Property

Three whole numbers that can be the side lengths of a right triangle are called a Pythagorean triple.

Examples

The numbers 3, 4, and 5 form a triple because $3^2 + 4^2 = 9 + 16 = 25$, which equals $5^2$.
The set 5, 12, and 13 is a triple since $5^2 + 12^2 = 25 + 144 = 169$, which is $13^2$.
Multiplying the (3, 4, 5) triple by 3 gives (9, 12, 15), which is also a triple: $9^2 + 12^2 = 81 + 144 = 225 = 15^2$.

Explanation

These are the VIP members of the right triangle club! Pythagorean triples are special sets of three whole numbers that fit perfectly into the $a^2 + b^2 = c^2$ formula, giving you a perfect right triangle without any messy decimals. Even cooler, if you multiply a triple by any whole number, you get a new triple!

Property

If the sum of the squares of two sides of a triangle equals the square of the third side, then the triangle is a right triangle.

Examples

A triangle with sides 8, 15, and 17 is a right triangle because $8^2 + 15^2 = 64 + 225 = 289$, and $17^2 = 289$.
Is a triangle with sides 7, 9, and 12 a right triangle? No, because $7^2 + 9^2 = 49 + 81 = 130$, which does not equal $12^2 = 144$.
To check if a corner is a right angle, you can measure 6 feet and 8 feet along the walls. The distance between those points must be 10 feet.

Explanation

This is the theorem's detective twin! Instead of starting with a right angle, you start with three side lengths and use the formula to check if you have a right triangle. If $a^2 + b^2 = c^2$ is true, you've found a 90-degree angle. It's the ultimate tool for builders and designers to confirm perfect corners.