A side that is an altitude

The idea

A side of a triangle is also an altitude when one of the triangle's angles is a right angle; triangles having one right angle are called right-angled triangles or right triangles.

Can a side be an altitude?

An altitude from A has to meet BC at 90°. Look at the two sides AB and AC. They both start at A.

Can the side AB itself be the altitude from A?

• Never, an altitude is always inside
• Yes, if the angle at B is 90°
• Yes, if AB = AC

Make a side the altitude

Drag A until the altitude from A is a side of the triangle. Then look at the other corners.

Right angle, side altitude

When the angle at B is 90°, the side BA is already at 90° to BC. So AB is the altitude from A, and in the same way CB is the altitude from C.

A triangle with one right angle is a right-angled triangle, or a right triangle.

A side of a triangle is also an altitude when one angle is a right angle. A triangle with one right angle is a right-angled triangle (right triangle): two of its three altitudes are its sides.

• Right angle at B | The altitude from it is
• from A, to BC | the side AB
• from C, to AB | the side CB
• from B, to AC | a segment inside the triangle

Notes

A side of a triangle is also an altitude when one angle is a right angle. A triangle with one right angle is a right-angled triangle (right triangle): two of its three altitudes are its sides.

Check yourself

In triangle PQR the angle at Q is 90°. How many of the three sides are also altitudes of the triangle?

Answer: 2

PQ is at 90° to QR, so PQ is the altitude from P. QR is at 90° to PQ, so QR is the altitude from R. That is two sides.

Which triangle has a side that is also an altitude?

In a triangle ABC the angle at B is 90°. What is the altitude from A to BC?

Why is AB the altitude from A when ∠B = 90°?

• The triangle with angles 60°, 60° and 60°. It has no right angle, so no side is at 90° to another side. All three altitudes fall inside.
• The triangle with angles 90°, 40° and 50° — correct. Yes! It has a right angle, so the two sides at the right angle are altitudes.
• The triangle with angles 100°, 40° and 40°. It has an obtuse angle, not a right angle. Its altitudes from the other corners fall outside.
• A segment inside the triangle, meeting BC between B and C. That is the altitude from B. Here AB is already at 90° to BC.
• A segment outside the triangle. It falls outside only when an angle at an end of the base is obtuse. Here ∠B is 90°.
• The side AB itself — correct. Yes! AB is already at 90° to BC and goes from A to BC.
• AB is the longest side. The length does not decide it. The 90° angle between AB and BC does.
• AB is the shortest side. The length does not decide it. The 90° angle between AB and BC does.
• AB goes from A to the line BC and is at 90° to BC — correct. Yes! That is exactly what an altitude is.