Finding the Hypotenuse Using the Pythagorean Theorem

Vocabulary and key concepts regarding the use of the Pythagorean theorem to calculate the hypotenuse of a right triangle.

Pythagorean Theorem

A mathematical theorem expressed as $a^2 + b^2 = c^2$ used to find the length of the hypotenuse in a right triangle when the lengths of the two legs are known.

Hypotenuse

The longest side of a right triangle, represented by the variable 'c', which is always opposite the right angle.

Legs

The two sides of a right triangle represented by the variables 'a' and 'b' whose order of assignment is interchangeable.

Right Triangle

A triangle characterized by having exactly one $90$-degree angle.

Solving for 'c'

A four-step process: 1. Substitute leg lengths for 'a' and 'b'; 2. Square 'a' and 'b'; 3. Add the squared values; 4. Take the square root of the sum.

Irrational Numbers

The result produced when the square root of the sum of $a^2 + b^2$ is not a whole number, often requiring the answer to be rounded to a specific decimal place.

Example 1 (Whole Number Result)

A calculation where legs are $8,m$ and $6,m$, resulting in $8^2 + 6^2 = 100$, where $\sqrt{100} = 10,m$.

Example 2 (Irrational Result)

A calculation where legs are $10,ft$ and $7,ft$, resulting in $10^2 + 7^2 = 149$, where $c = \sqrt{149} \approx 12.21,ft$ when rounded to the hundredths place.