Unit 11: Triangles

Tear the corners off

Draw any triangle, tear off its three corners, and lay them side by side.

They always make a straight line. Every time, for every triangle anyone has ever drawn.

So the three angles of a triangle add to 180°.

Triangles by their sides

Equilateral — all three sides equal, and so all three angles are 60°.

Isosceles — two sides equal, and the two angles opposite them are equal too.

Scalene — all different.

That middle one does most of the work: equal sides mean equal angles, and it goes both ways.

Two squares that fit inside a third

Take a right-angled triangle. Draw a square on each of the two short sides, and one on the long side.

The two small squares, added together, have exactly the area of the big one.

Written in letters, with c the long side: a^2 + b^2 = c^2. The long side, opposite the right angle, is the hypotenuse.

Going backwards

If you know the long side, subtract instead of adding.

A ladder 13 m long leans with its foot 5 m from the wall. How high does it reach?

13^2 - 5^2 = 169 - 25 = 144, and sqrt(144) = 12. Twelve metres.

The rule runs both ways

In an isosceles triangle the two equal sides sit opposite two equal angles.

And the reverse is true too: if two angles are equal, the sides opposite them must be equal as well.

So spotting either one gives you the other free. That is why isosceles triangles do so much work in angle chasing.

Which angle is the odd one out?

Careful. "An isosceles triangle has an angle of 40°" does not say *which* angle.

If 40° is the top, the base angles are 70° each.
If 40° is a base angle, the other base angle is also 40° and the top is 100°.

Two different triangles, both correct. A good question says which.

When Pythagoras comes out whole

Most right-angled triangles have an ugly hypotenuse. A few are beautifully tidy, and those turn up again and again:

3, 4, 5 · 5, 12, 13 · 8, 15, 17 · 7, 24, 25

Learn those four and you will recognise half the questions you meet without doing any arithmetic at all.

Every triple has infinitely many cousins

Double 3, 4, 5 and you get 6, 8, 10 — also right-angled, because the whole triangle just got twice as big.

Triple it: 9, 12, 15. Times ten: 30, 40, 50.

So if the numbers in a question look like a multiple of a triple, divide through and check the small version first.

Running the test backwards

Pythagoras also works as a test. If a^2 + b^2 really does equal c^2, the triangle must be right-angled — no measuring needed.

Sides 9, 40, 41? 81 + 1600 = 1681, and 41^2 = 1681. Right-angled.

Sides 4, 5, 6? 16 + 25 = 41, but 36 is not 41. Not right-angled.

The short way is always shorter

Try to build a triangle with sides 2, 3 and 10. You cannot — the two short sides together do not reach.

Any two sides must add to more than the third. Walking straight there beats walking via a corner, always.

Check only the two shortest against the longest. If that works, the others follow automatically.

How big can the third side be?

Two sides are 7 and 10. What can the third be?

It must be less than 7 + 10 = 17, or the two shorts cannot reach.
And more than 10 - 7 = 3, or the same problem appears from the other side.

So the third side lies strictly between 3 and 17.

Biggest side, biggest angle

One more fact worth carrying: in any triangle the largest angle sits opposite the longest side.

So in a triangle with sides 5, 7 and 9, the biggest angle is opposite the 9.

It matches the isosceles rule — equal sides give equal angles — extended to the case where nothing is equal.

Three special lines

A median joins a corner to the middle of the opposite side.

An altitude drops from a corner straight down onto the opposite side, at a right angle. That is the height used in the area formula.

A bisector cuts an angle exactly in half.

In an equilateral triangle all three are the same line. In a scrappy one they are three different lines.

A median halves the area

A median splits a triangle into two smaller triangles with the same base length and the same height.

So they have equal areas — every time, no matter how lopsided the triangle looks.

That is the tool for a great many "find the shaded area" problems.

When you have no height

Every area formula so far has wanted a height. But a triangle with sides 5, 6 and 7 has no obvious height at all.

Heron of Alexandria found a way round it. First take the semi-perimeter — half the way round:

s = (a + b + c)/2

For 5, 6, 7 that is 9. Then one formula finishes the job.

Heron’s formula

Take s, then subtract each side from it in turn, multiply all four numbers together, and square-root the lot:

area = sqrt(s(s - a)(s - b)(s - c))

For 5, 6, 7: sqrt(9 × 4 × 3 × 2) = sqrt(216), which is about 14.7.

No height, no angles, no drawing. Three side lengths are enough.

Congruent and similar

Two triangles are congruent if they are identical — same angles *and* same sizes. One could be picked up and placed exactly on the other.

They are similar if they are the same shape but different sizes, like a photo and its enlargement. Same angles, sides all scaled by the same amount.

Similar triangles are the workhorse: they let you measure things you cannot reach.

Two angles are enough

To check two triangles are similar, you do not need all three angles — if two match, the third must as well, because they all add to 180°.

And once they are similar, every pair of matching sides is in the same ratio. That single fact solves an enormous number of problems.

Scaling areas

If a triangle is scaled to be 3 times as long, its area goes up by 3^2 = 9 times, not 3.

Every dimension grew, so area — which is two dimensions multiplied — grows by the scale factor squared.

The same reason a 12-inch pizza beats a 6-inch one by four, back in unit 9.