Understanding the Pythagorean Theorem

The Pythagorean Theorem is a fundamental rule in geometry that applies only to right triangles (triangles with one 90-degree angle). It states that the square of the longest side, called the hypotenuse, is equal to the sum of the squares of the other two sides, called the legs.

If $a$ and $b$ are the lengths of the legs, and $c$ is the length of the hypotenuse, the formula is: $a^2 + b^2 = c^2$

Finding the Hypotenuse

When you know the lengths of both legs, you can use the theorem to find the hypotenuse.

Example: A right triangle has legs of length $3$ and $4$. Find the hypotenuse.

1. Plug the known values into the formula: $3^2 + 4^2 = c^2$
2. Square the numbers: $9 + 16 = c^2$
3. Add them together: $25 = c^2$
4. Take the square root of both sides: $c = 5$

The hypotenuse is $5$.

Finding a Missing Leg

If you know the length of the hypotenuse and one leg, you can work backward to find the missing leg.

Example: The hypotenuse is $13$ and one leg is $5$. Find the other leg.

1. Set up the equation: $5^2 + b^2 = 13^2$
2. Square the known numbers: $25 + b^2 = 169$
3. Subtract $25$ from both sides to isolate $b^2$: $b^2 = 144$
4. Take the square root: $b = 12$

The missing leg is $12$.

Checking if a Triangle is a Right Triangle

You can also use the theorem in reverse (called the converse of the Pythagorean Theorem). If the square of the longest side equals the sum of the squares of the two shorter sides, the triangle must be a right triangle.

Example: Is a triangle with sides $5$, $12$, and $13$ a right triangle?

1. Identify the longest side (the potential hypotenuse), which is $13$.
2. Check if $a^2 + b^2 = c^2$ by squaring and adding the two shorter sides: $5^2 + 12^2 = 25 + 144 = 169$
3. Square the longest side: $13^2 = 169$

Since $169 = 169$, the equation holds true. Therefore, it is a right triangle.