Hypotenuse

The idea

In a right-angled triangle, the side opposite to the right angle is called the hypotenuse.

A special side

A right-angled triangle has a right angle, the little square corner. One of its three sides has a special name. Which side do you think it is?

Which side of a right-angled triangle is special?

• The side across from the right angle
• The side at the bottom
• A side that touches the right angle

Find the special side

Four right-angled triangles lie in different positions. In each one, tap the side that is across from the right angle.

The hypotenuse

The side that is across from the right angle is the hypotenuse. It is not always at the bottom. It does not touch the right angle. It was also the longest side of every triangle.

In a right-angled triangle, the side opposite to the right angle is called the hypotenuse.

• Side | Touches the right angle?
• the two legs | yes
• the hypotenuse | no, it is across from it

Notes

In a right-angled triangle, the side opposite to the right angle is called the hypotenuse.

Check yourself

In ∆PQR the right angle is at Q. Which side is the hypotenuse?

A right-angled triangle is turned upside down. Which side is now its hypotenuse?

A right-angled triangle has sides of 6 cm, 8 cm and 10 cm. How long is its hypotenuse, in cm?

Answer: 10

In a right-angled triangle the hypotenuse is the longest side, so it is 10 cm. The legs are 6 cm and 8 cm.

Can a triangle have two hypotenuses?

• PR — correct. Yes! PR is the side across from Q, the corner with the right angle.
• PQ. PQ touches the right angle at Q. The hypotenuse is across from Q.
• QR. QR touches the right angle at Q too. The hypotenuse is across from Q.
• The same side as before: the one across from the right angle — correct. Yes! Turning does not change which side is across from the right angle.
• The side at the bottom. The bottom changes when we turn the triangle. The hypotenuse does not.
• The side at the top. The top changes when we turn the triangle. The hypotenuse does not.
• Yes, if it has two long sides. The hypotenuse is not chosen by length alone. It is across from the right angle.
• No. A triangle has at most one right angle, so only one side is across from it. — correct. Yes! Two right angles would make 180°, leaving nothing for the third angle.
• Yes, every side is a hypotenuse in turn. Only the side across from the right angle is the hypotenuse.