Right Triangle

A right angle has a value of 90 degrees ($90^\circ$). A right triangle is a triangle in which one angle is a right angle. The relation between the sides and angles of a right triangle is the basis for trigonometry.

The side opposite the right angle is called the hypotenuse (side $c$ in the figure). The sides adjacent to the right angle are called legs (sides $a$ and $b$). Side $a$ may be identified as the side adjacent to angle $B$ and opposed to (or opposite) angle $A$. Side $b$ is the side adjacent to angle $A$ and opposed to angle $B$.

Right triangle

The Pythagorean Theorem can be used to find the value of a missing side length in a right triangle.

If the lengths of all three sides of a right triangle are whole numbers, the triangle is said to be a Pythagorean triangle and its side lengths are collectively known as a Pythagorean triple.

The Pythagorean Theorem

The Pythagorean Theorem, also known as Pythagoras' Theorem, is a fundamental relation in Euclidean geometry. It defines the relationship among the three sides of a right triangle. It states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. The theorem can be written as an equation relating the lengths of the sides $a$, $b$ and $c$, often called the "Pythagorean equation":^[1]

${\displaystyle a^{2}+b^{2}=c^{2}} $

In this equation, $c$ represents the length of the hypotenuse and $a$ and $b$ the lengths of the triangle's other two sides.

Although it is often said that the knowledge of the theorem predates him,^[2] the theorem is named after the ancient Greek mathematician Pythagoras (c. 570 – c. 495 BC). He is credited with its first recorded proof.

The Pythagorean Theorem

The sum of the areas of the two squares on the legs ($a$ and $b$) is equal to the area of the square on the hypotenuse ($c$).  The formula is $a^2+b^2=c^2$.

Finding a Missing Side Length

Example 1:  A right triangle has a side length of $10$ feet, and a hypotenuse length of $20$ feet.  Find the other side length.  (round to the nearest tenth of a foot)

Substitute $a=10$ and $c=20$ into the Pythagorean Theorem and solve for $b$.

$\displaystyle{ \begin{aligned} a^{2}+b^{2} &=c^{2} \\ (10)^2+b^2 &=(20)^2 \\ 100+b^2 &=400 \\ b^2 &=300 \\ \sqrt{b^2} &=\sqrt{300} \\ b &=17.3 ~\mathrm{feet} \end{aligned} }$

Example 2:  A right triangle has side lengths $3$ cm and $4$ cm.  Find the length of the hypotenuse.

Substitute $a=3$ and $b=4$ into the Pythagorean Theorem and solve for $c$.

$\displaystyle{ \begin{aligned} a^{2}+b^{2} &=c^{2} \\ 3^2+4^2 &=c^2 \\ 9+16 &=c^2 \\ 25 &=c^2\\ c^2 &=25 \\ \sqrt{c^2} &=\sqrt{25} \\ c &=5~\mathrm{cm} \end{aligned} }$