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Hypotenuse: Definition & Formula

Hypotenuse: Definition & Formula
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  • 0:00 What Is a Hypotenuse?
  • 0:21 Mathematical Uses
  • 2:31 Practical Applications
  • 4:14 Lesson Summary
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Lesson Transcript
Instructor: Tim Brown
The term hypotenuse refers to the longest side of a right triangle and has many properties important to both geometry and trigonometry. This lesson will describe the hypotenuse and give insight into its importance.

What Is a Hypotenuse?

In geometry, a hypotenuse is the longest side of a right triangle. It's also the side opposite the right angle. The word hypotenuse means 'length under' or 'stretch under.' Its earliest use was thought to be by Plato in approximately 400 BC.

Mathematical Uses

The hypotenuse figures prominently in many applications in both geometry and trigonometry. The one most people are aware of is the Pythagorean Theorem. The length of the hypotenuse is found by using the Pythagorean Theorum, a^2 + b^2 = c^2. Let's look at an example.

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Plug in the values for a and b into the Pythagorean Theorem.

10^2 + 12^2 = c^2

Then, solve.

100 + 144 = c^2

244 = c^2

c = √(244)

c = 15.6 ft.

The theorem can also be used not only to find the length of either side when the hypotenuse and one side are known. Finding the measures of the angles of a triangle also can be done if you know the length of the hypotenuse and at least one other side of the triangle. Or, you can find the length of a side if you know the angle measure and at least one other side.

The mnemonic SOHCAHTOA is helpful for remembering what goes with what.

  • SOH = the sine of an angle is equal to the side opposite the angle divided by the hypotenuse
  • CAH = the cosine of an angle is equal to the side adjacent to the angle divided by the hypotenuse
  • TOA = the tangent of an angle is equal to the side opposite the angle divided by the side adjacent to the angle

With these properties, you can solve almost any problem related to finding either a side length or angle measure of a right triangle. Let's look at an example. Find x in this triangle:

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Because we know the angle adjacent to side x and the hypotenuse, the equation we should use is CAH as it uses the adjacent side and hypotenuse. Therefore, our equation will be cos60 = x/13. The cos of 60 is 0.5, which makes the equation: 0.5 = x/13. Solve for x to get x = (0.5) * (13) = 6.5. So, the length of side x is 6.5 cm.

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