Sides of a right triangle

Measuring what you cannot reach

How tall is the clock tower in your town? You cannot climb it with a measuring tape. But you can stand some distance away, measure that distance, and measure the angle up to the top. A right triangle then gives the height.

This is what trigonometry does. The word comes from Greek: tri (three), gon (sides) and metron (measure). It is the study of how the sides and the angles of a triangle depend on each other. Surveyors, sailors, engineers and astronomers use it every day.

In this course every triangle is a right triangle: one angle is $90°$. The other two angles are acute, each less than $90°$, and together they make $90°$.

The hypotenuse

The side opposite the right angle is the hypotenuse. It is always the longest side. Pythagoras' theorem links it to the other two sides:

A worked example. In triangle $PQR$ the right angle is at $Q$, $PQ = 9\ \text{cm}$ and $QR = 12\ \text{cm}$. Find $PR$.

1. Spot the hypotenuse. It faces the right angle at $Q$, so it is $PR$.
2. Use Pythagoras. $PR^2 = 9^2 + 12^2 = 81 + 144 = 225$.
3. Take the root. $PR = 15\ \text{cm}$.

Opposite and adjacent

Now pick one of the acute angles, say angle $A$ in triangle $ABC$ with the right angle at $B$.

• The opposite side is the side facing angle $A$. It does not touch $A$. Here it is $BC$.
• The adjacent side is the side next to angle $A$ that is not the hypotenuse. Here it is $AB$.
• The hypotenuse $AC$ also touches angle $A$, but it keeps its own name.

The names depend on the angle you look from. Seen from angle $C$, the side $AB$ is now opposite, and $BC$ is adjacent. The hypotenuse never changes.

A habit that saves marks

Before any trigonometry question, do three things.

1. Find the right angle. Mark the hypotenuse opposite it.
2. Find the angle in the question. Mark the side that faces it as opposite.
3. Name the last side adjacent. It touches the angle and is not the hypotenuse.

A worked example. In triangle $XYZ$ the right angle is at $Z$. For angle $X$, which side is which?

1. Hypotenuse. It faces $Z$, so it is $XY$.
2. Opposite. It faces $X$, so it is $YZ$.
3. Adjacent. The remaining side, $XZ$.

Practise with new numbers

Try it in Python

This program finds the hypotenuse of the triangle with sides $9\ \text{cm}$ and $12\ \text{cm}$, then names the other two sides from each acute angle.

Common mistakes

• Calling the hypotenuse adjacent. It touches the angle, but it is always called the hypotenuse.
• Naming sides without an angle. "The opposite side" means nothing until you say which angle.
• Looking from the right angle. Opposite and adjacent are named from an acute angle only.
• Adding sides instead of squares. The hypotenuse is $\sqrt{a^2 + b^2}$, not $a + b$.

Remember these

• The hypotenuse faces the right angle and is the longest side.
• The opposite side faces the chosen angle; the adjacent side touches it and is not the hypotenuse.
• Change the angle and opposite and adjacent swap.
• Pythagoras: $\text{hypotenuse}^2 = \text{side}_1^2 + \text{side}_2^2$.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. In triangle $DEF$ the right angle is at $E$. Which side is the hypotenuse?

2. In the same triangle, which side is opposite angle $D$?

3. A right triangle has shorter sides $7\ \text{cm}$ and $24\ \text{cm}$. How long is the hypotenuse?

4. Rani says the adjacent side of angle $A$ is always the bottom side of the drawing. Is she right?

5. Can the opposite side of an acute angle be longer than the hypotenuse?