What an altitude is, and how to construct one

0:00A pole. Five metres of it, standing straight up. How tall is it? Five metres. Obviously. Now lean it over. Twenty degrees off upright. It is still five metres of pole. Not a centimetre of it went anywhere. But it now stands four point seven metres tall. Lean it to thirty-five degrees and it stands four point one. So height was never a property of the pole. It was a measurement of the pole against the ground.

0:32Which raises an awkward question for a triangle. A triangle does not come with a ground. So you have to give it one. Here is a triangle. Choose a side, any side, and call it the base. This one, B C, fourteen centimetres. The other two are fifteen and thirteen, so nothing here is symmetric. Now the question means something. How high is the corner A above that base? Go from A to the base, and meet it at a right angle.

1:06Mark the right angle. Call the point where it lands D. That segment, A down to D, is twelve centimetres. Twelve is the height of A above B C. That is what a height in a triangle is. The segment itself has a name. It is an altitude. From a corner, to the opposite side, meeting that side at a right angle. That is the whole definition. Read it again, though, for what it does not say.

1:37It does not say the middle of anything. It does not say straight down the page. Down the page only means something if the base happens to be lying flat, and nothing says it must. Tilt the whole triangle and the altitude tilts with it. What survives is the right angle. And it says nothing whatsoever about staying inside the triangle. Hold on to that one. Two things look true in that picture and are not.

2:08The foot, D, sits nine centimetres from B and five from C. The middle of a fourteen centimetre side is at seven. The altitude misses it by two whole centimetres. It does not cut the base in half. And look at the corner it came from. It splits the angle at A into thirty-seven degrees and twenty-three. It does not cut that in half either. There is one case where it does both at once, and that is when the two sides running down from A are the same length.

2:42Out of every triangle whose angles are whole numbers of degrees, fifteen thousand nine hundred and thirty-one of them, that happens in eighty-nine. Now, A was not special. It was just the corner we happened to ask about. B has one too. Stand the triangle on C A instead, and drop the perpendicular from B. That one is twelve point nine centimetres. And C has one, down onto A B. Eleven point two.

3:15Each starts at a corner, ends on the side facing it, and meets it square on. So one triangle, three altitudes. Twelve, twelve point nine, and eleven point two. Three different numbers, out of one shape that never changed. Which makes the phrase everybody says, the height of the triangle, quietly meaningless. It is shorthand, and it is short for something longer. The height of the triangle means the height for whichever side you are calling the base.

3:50Rest it on the fourteen and the height is twelve. On the thirteen, twelve point nine. On the fifteen, eleven point two. Which is worth staring at, because there is a pattern in it. The longest side got the shortest height. So multiply each height by the side it is standing on. Twelve times fourteen is a hundred and sixty-eight. Twelve point nine times thirteen is a hundred and sixty-eight.

4:20Eleven point two times fifteen is a hundred and sixty-eight. The same number, three times over. The base and the height trade against each other exactly. Now the case that catches everybody. This triangle has a very wide corner at B. A hundred and forty degrees, with sides of seven and four running from it. Take B C as the base, and ask how high A is above it. Set off from A towards the base at a right angle, and there is nothing there.

4:55A is not above B C at all. It is off to the side of it. So extend the base line. Carry it on past B. Five point four centimetres past B, and now there is something to be perpendicular to. The altitude from A comes out at four point five centimetres. It is drawn dashed, because it is not inside the triangle. Two things about that, and both of them matter.

5:25First, the side B C is still four centimetres. It was four before and it is four now. The line through B and C got longer. The side did not. That extension is a drawing aid. Not a fourth side, and not part of the shape. Second, this is not some rare curiosity. Of those fifteen thousand nine hundred and thirty-one triangles, eleven thousand seven hundred and forty-eight have two altitudes falling outside.

5:57That is nearly three quarters of them. And it is always exactly two. Never one, never all three. The two that go outside come from the two sharp corners. The one from the wide corner lands inside, every single time. You can make one of these with no instruments at all. Cut a triangle out of paper. Fold it so the crease runs through the top corner and the base lands back on itself.

6:28Open it out. The crease is the altitude, and here is why it has to be. A fold is a mirror. Whatever is on one side of the crease is the reflection of the other. So the two angles the crease makes with the base are mirror images. They are equal. And the base is straight, so between them they fill a straight line. A hundred and eighty degrees. Two equal angles adding to a hundred and eighty are ninety each. The crease is perpendicular, and no protractor went near it.

7:04I tried every crease direction, a tenth of a degree apart. Exactly one lays the base back onto itself. Ninety. Which brings us to why you should not simply eyeball it. Take that twelve centimetre altitude and draw it six degrees off square. Eighty-four instead of ninety. Measure what you drew. Twelve point zero seven centimetres. Seven hundredths of a centimetre too long. You would never catch that with a ruler.

7:37But look where it landed. Its foot is a centimetre and a quarter from where it belonged. Ten degrees off and the foot is more than two centimetres out. Fifteen degrees off and it is over three. The length barely notices. The position gives it away completely. So the right angle is the one thing you cannot afford to guess at. Which means using something that already has one. A set square. A drawing triangle with a right angle built into the corner.

8:12Three moves. Here is a triangle to do them on: base five centimetres, one side five and one six. First, align. Lay the ruler along the base and stand the set square on it, one arm of its right angle flat against the ruler. Second, slide. Push it along the ruler, keeping that arm pressed flat. The upright edge stays square to the base the whole way along. Sliding cannot tilt it.

8:42Keep going until it reaches the corner you are working from. Here that is three point six centimetres along. Third, draw. Run the pencil down the upright edge, from the corner to the base. Four point eight centimetres, at a right angle, landing one point four from the far end. Slid, not guessed. One last shape, where some of the work is already done for you. A triangle with a right angle in it, at B, with A B three centimetres and B C four.

9:17Take B C as the base, and ask for the altitude from A. It is already drawn. A B meets B C at a right angle, and that is the entire requirement. Now take A B as the base and ask for the altitude from C. It is C B, four centimetres. Two of the three altitudes are sides of the triangle itself. Only the third has to be drawn, from B to the long side. Two point four centimetres. And two point four times five is twelve, just as three times four is.

9:55That shortcut belongs to right angles alone. A side doubles as an altitude when the triangle has one, never otherwise.