What do you understand by the Term hypotenuse?
In Mathematics, the ‘hypotenuse’ has its origin from a Greek word ‘hypotenuse’, which means ‘stretching under’.
In a right angled triangle, the side opposite to the 90 degree angle is known as the hypotenuse. In other words, it can be said that the longest side in a right triangle is known as its hypotenuse.
As mentioned, the hypotenuse is the longest side of a right triangle compared to its legs which is also the length of the base and the perpendicular or the height.
The hypotenuse also lies on the side opposite to the right angle, which is the greatest angle of all the three angles in a right triangle. You can better understand the hypotenuse in a right triangle by knowing about the right-angled theorem, or, by applying the Pythagorean theorem.
Pythagorean Theorem
Pythagorean theorem states that the sum of squares of the base and height of a right triangle is equal to the square of hypotenuse.
hypotenuse\(^2\) = perpendicular\(^2\) + base\(^2\)
h\(^2\) = p\(^2\) + b\(^2\)
h = \(\sqrt{p^2~+~b^2}\)
We can also write,
p = \(\sqrt{h^2~-~b^2}\)
b = \(\sqrt{h^2~-~p^2}\)
From the above formulae, you can calculate the hypotenuse, perpendicular, and base if any of the other two parameters are given.
Solved Length of Hypotenuse Examples
Example 1: Find the hypotenuse of the right triangle having base 5 meters and perpendicular height 12 meters.
Solution:
As stated in the question,
Base of triangle, b = 5 m,
Height of triangle, p = 12 m,
Hypotenuse, h = ?
h = \(\sqrt{p^2~+~b^2}\) [Apply formula]
= \(\sqrt{12^2~+~5^2}\) [Substitute the value]
= \(\sqrt{144~+~25}\) [Find the squares of the number]
= \(\sqrt{169}\) [Add]
= 13 m [Take the positive square root]
So, the hypotenuse of the triangle is 13 meters.
Example 2: Sam wants to measure the height of a pole which is 10 meters away from his foot and the slant height of the pole from his foot is 13 meters.
Solution:
As stated in the question,
Slant height of the pole, h = 13 m,
Base, b = 10 m
Height of the pole, p = \(\sqrt{h^2~-~b^2}\) [Apply formula]
= \(\sqrt{13^2~-~10^2}\) [Substitute the value]
= \(\sqrt{169~-~100}\) [Find the squares of the number]
= \(\sqrt{69}\) [Subtract]
= 8.306 m [Take the positive square root]
So, the height of the pole is 8.306 meters.
Example 3: Find the base of the triangle that has a hypotenuse of 15 centimeters and the height of the triangle is 5 centimeters.
Solution:
As stated in the question,
Perpendicular height(p) = 5 cm,
Hypotenuse(h) = 15 cm,
Base(b) = ?.
b = \(\sqrt{h^2~-~p^2}\) [Apply formula]
= \(\sqrt{15^2~-~5^2}\) [Substitute the value]
= \(\sqrt{225~~-~25}\) [Find the squares of the number]
= \(\sqrt{200}\) [Subtract]
= \(\sqrt{100~\times~2}\) [Factors of 200]
= 10\(\sqrt{2}\) cm [Take the positive square root of 100]
So, the base of the triangle is 10\(\sqrt{2}\) centimeters.
Yes, the hypotenuse is the longest side of a right triangle. It is opposite to the 90 degree angle in a right triangle. This is the largest angle of any right triangle. The side opposite to the greatest angle is the longest as per the Pythagorean theorem.
No, a right triangle cannot have an obtuse angle because the sum of all the angles of a triangle is 180 degrees. In the right triangle one angle is 90 degrees, which implies that the sum of the other two angles is 90 degrees.
No, a triangle can not have two right angles. The sum of all three angles of a triangle is 180 degrees. So, if a triangle has two right angles, then the third angle will measure 0 degrees, which is not possible.