Hypotenuse

In mathematics, the term "hypotenuse" is derived from the Greek hypoteinousa, which means "stretching under." The term is used in geometry to express the longest side of a right-angled triangle, or the side that is opposite the right angle.

Hypotenuse definition

The hypotenuse is a term specifically related to right triangles, which are triangles that have one angle measuring exactly 90 degrees. In a right triangle, the hypotenuse is the side opposite the right angle and is the longest side of the triangle. The term "hypotenuse" is not applicable to other types of triangles, such as acute or obtuse triangles.

In acute triangles, all angles are less than 90 degrees. In obtuse triangles, one angle is greater than 90 degrees. These triangles do not have a right angle, and therefore, they do not have a hypotenuse. Instead, they have three sides, with the longest side being opposite the largest angle.

The Pythagorean theorem

The hypotenuse plays a crucial role in the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. The formula for the Pythagorean theorem is:

${\mathrm{Hypotenuse}}^2={\mathrm{Base}}^2+{\mathrm{Perpendicular}}^2$

Hypotenuse formula

The formula above can be used to find the hypotenuse by taking the square root of the sum of the squares of the base and the perpendicular of a right triangle. which now gives us:

$\mathrm{Hypotenuse}=\sqrt{{\mathrm{Base}}^2+{\mathrm{Perpendicular}}^2}$

This can be made more simple (and standard) by rewriting it as $a^2+b^2=c^2$ , where a is the base, $b$ is the perpendicular, and $c$ is the hypotenuse. To solve for the length of the hypotenuse or either of the sides, simply solve the equation with at least 2 of the side lengths known.

Example 1

Solve for the hypotenuse of a right-angled triangle with a 12 ft. base and a 10 ft. perpendicular.

${12}^2+{10}^2=c^2$

Solve.

$144+100=c^2$

$244=c^2$

$c=\sqrt{244}$

$c=15.6\mathrm{ft}$ .

Practice using the Pythagorean formula

a. Solve for the hypotenuse of a right-angled triangle with $a$ 3 in. base and $a$ 4 in. perpendicular.

$3^2+4^2=5^2$

$9+16=c^2$

$25=c^2$

$c=\sqrt{25}$

$c=5\mathrm{in}$ .

b. Solve for the hypotenuse of a right-angled triangle with $a$ 7 in. base and $a$ 24 in. perpendicular.

$7^2+{24}^2=c^2$

$49+576=c^2$

$625=c^2$

$c=\sqrt{625}$

$c=25\mathrm{in}$ .

Topics related to the Hypotenuse

30-60-90 Triangles

Triangles

Area of a Triangle

Flashcards covering the Hypotenuse

Basic Geometry Flashcards

Common Core: High School - Geometry Flashcards

Practice tests covering the Hypotenuse

Common Core: High School - Geometry Diagnostic Tests

Intermediate Geometry Diagnostic Tests

Get help learning about the hypotenuse

Finding which line is the hypotenuse on a right-angled triangle is not that difficult, but it is just the beginning. Following the Pythagorean theorem, also called the hypotenuse theorem, finding the length of the hypotenuse based on the length of the base and perpendicular can get a little bit tricky. Working with a tutor can give your student a leg up in learning to use this theorem on the go, which is essential if they are taking intermediate or high school geometry or trigonometry.

A private tutor can work 1-on-1 with your student to make sure they understand the mathematical fundamentals underlying the theorem. They can also make sure your student is able to perform the functions necessary to find the right answers. Get in touch with the Educational Directors at Varsity Tutors today to see how tutoring can help your student and get signed up today.