Dropping straight down

The idea

An altitude is the perpendicular segment from a vertex to the opposite side, extended if needed; its length is that vertex's height, and a triangle's height generally means the altitude to the side taken as base.

How tall is a triangle?

A triangle ABC stands on its side BC. You want to measure how tall it is, from the corner A.

How would you measure it?

• Along the slanting side AB
• Straight down from A, meeting BC at a right angle
• From A to the middle of BC

Drop the altitude

Tap a corner. A segment is dropped from it to the opposite side. Then drag A far to one side.

Altitude and height

The segment dropped from a corner at a right angle to the opposite side is an altitude. Its length is the height.

If the foot falls outside the triangle, the side is extended to meet it. A triangle has three altitudes, AD, BE and CF, one from each corner. “The height” means the altitude to the side you take as the base.

An altitude is the perpendicular segment from a corner to the opposite side, extended if needed. Its length is the height. “The height of a triangle” means the altitude to the side taken as the base.

• Word | Meaning
• Altitude AD | the segment from A to the line BC, at 90° to BC
• Height | the length of the altitude
• Base | the side the altitude falls on

Notes

An altitude is the perpendicular segment from a corner to the opposite side, extended if needed. Its length is the height. “The height of a triangle” means the altitude to the side taken as the base.

Check yourself

Which segment is the altitude from A in triangle ABC?

How many altitudes does a triangle have?

Answer: 3

There is one altitude from each corner: AD, BE and CF. A triangle has three altitudes.

In a triangle the angle at B is 110°. Where does the altitude from A meet the line BC?

The foot of the altitude from A falls outside the side BC. What must be true?

• The segment from A to the line BC that is at 90° to BC — correct. Yes! An altitude is the perpendicular from the corner to the opposite side.
• The segment from A to the middle of BC. The middle of BC is not asked for. An altitude must meet the opposite side at 90°.
• The side AB. AB slants across BC. An altitude must make 90° with the opposite side.
• On BC extended beyond B, outside the triangle — correct. Yes! With an obtuse angle at B, the side CB has to be extended to meet the altitude from A.
• Between B and C, inside the triangle. That happens when the angles at B and C are both smaller than 90°. Here ∠B is obtuse.
• There is no altitude from A. Every corner has an altitude. Here it just falls outside the triangle.
• AB and AC are equal. Equal sides AB and AC put the foot in the middle of BC, inside the triangle.
• The triangle is very tall. The height does not decide it. What matters is whether an angle at B or C is obtuse.
• The angle at B or the angle at C is obtuse — correct. Yes! The side has to be extended only when one of the angles at its ends is bigger than 90°.