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The Pythagorean theorem is a cornerstone of geometry, providing a powerful relationship between the sides of a right triangle. A right triangle is defined as a triangle with one interior angle measuring exactly.

The Formula

The theorem states that for any right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

The formula is expressed as:

Where:

• $a$ and $b$ are the lengths of the two shorter sides, called the legs.

• $c$ is the length of the longest side, called the hypotenuse.

How to Use It

The theorem can be used to find the length of any side of a right triangle as long as you know the lengths of the other two sides.

Example 1: Finding the Hypotenuse ($c$)

Let's say you have a right triangle with legs of length 3 and 4. You want to find the length of the hypotenuse.

1. Write the formula: $a^2+b^2=c^2$
   

2. Substitute the known values: $3^2+4^2=c^2$
   

3. Solve for $c$: $9+16=c^2$ $25=c^2$ $$\begin{gathered} c= \\ sqrt25 \end{gathered}$$ $c=5$
   

The length of the hypotenuse is 5.

Example 2: Finding a Leg ($a$ or $b$)

Now, let's say you know the hypotenuse is 13 and one leg is 5. You want to find the length of the other leg.

1. Write the formula: $a^2+b^2=c^2$
   

2. Substitute the known values: $5^2+b^2=13^2$
   

3. Solve for $b$: $25+b^2=169$ $b^2=169-25$ $b^2=144$ $$\begin{gathered} b= \\ sqrt144 \end{gathered}$$ $b=12$
   

The length of the missing leg is 12.