Which quadrilaterals you can build by joining two triangles

0:00Cut out a triangle. Then cut out a second one exactly like it. Now press them together along an edge — but not any two edges. The two have to be the same length, or they will not meet. That one rule is doing more work than it looks like it is doing. So a triangle with three different sides offers three choices, not nine: a 6 to a 6, a 9 to a 9, a 12 to a 12.

0:29So already the question has a shape to it. Which edge do you join along? And there is a second question underneath it, which is easy to miss. Between them, those two questions decide everything about the figure you end up with. Watch what happens to the edge you joined along. Before you press, it is an edge of each triangle. After, it is an edge of nothing. It has gone inside. It runs from one corner of the new figure to the opposite corner, and that is what a diagonal is.

1:06The four sides of the new shape are the four edges you did not join: two from one cutout, two from the other. This is the part that gets misremembered. Count that edge as a fifth side and nothing you say afterwards comes out right. Draw it faint the moment the join is made, and you will not. So joining two triangles hands you a quadrilateral with its diagonal already drawn in.

1:36Here is the second question. Hold the first cutout still, and take the twin. You can turn it round — half a turn about the middle of the joining edge — and lay it down. Or turn it over, flipping it across that edge like closing a book. Both fit. The joined edge is the same length either way. But they are not the same placement, and the shapes they give are not the same shape.

2:03Which edge, and which way. Two questions, three answers to the first and two to the second. Six builds from any triangle at all, and nothing else to choose. Turn the twin round, and something is forced immediately. Each side faces the same edge of the cutout, arriving the other way up. So the facing sides come in equal pairs, and run parallel. That is a parallelogram, every time, without measuring anything.

2:36Turn the twin over instead and each edge lands next to its own copy rather than opposite it. The equal sides are neighbours. That is a kite. Twenty-seven builds were put to both questions, then fifty-one more. Every turned-round one gave equal facing sides, every turned-over one equal neighbours. Ask each placement the other question and not one answers yes. Start with the easiest cutout: an equilateral triangle, 8 on every side.

3:13Every edge is the same, so which one you join along cannot matter. It turns out which way you place the twin does not either. Six builds, and all six are the same figure. Its four sides are all 8, because each of them is an edge of a cutout and every one of those was 8. Four equal sides. That is a rhombus, and you can say so without picking up a ruler.

3:40Now the corner it is tempting to cut. It is not a square. Its corners are 60 and 120 — one arriving whole, two meeting to make the other. Four equal sides, and no square corner anywhere. The diagonal you drew faintly is worth measuring. It is the edge you joined along, so it is 8, like every side. The other one, between the two far corners, is about 13.9 — longer than any side of the figure.

4:15So the two diagonals are nothing like each other, though the four sides are identical. That is worth holding on to. Equal sides say nothing at all about the diagonals. And notice which one you got for free: the short one, because it was an edge of the cutout before it was anything else. Every figure in this video arrives with one diagonal already known, and it is always the edge you joined along.

4:46Change the cutout. Two sides of 8 and one of 6 — an isosceles triangle. Join the pair along the odd edge, the 6, and turn the twin round. The four sides are the four 8s, so it is a rhombus again. The 6 runs through the middle as the diagonal. Its corners are 44 and 136 this time, not 60 and 120. Same four equal sides, different figure. Now turn the twin over across that same 6 instead, and watch carefully.

5:23You get the identical shape. Not a similar one — the same one. The cutout is already symmetric about that edge, so both placements put the far corner in the same spot. Join the same pair along one of the 8s instead. Turned round, the sides come out 6, 8, 6, 8 — a parallelogram, with corners of 112 and 68. Turned over, the same lengths rearrange: 6, 8, 8, 6. The equal pairs are neighbours, so it is a kite.

5:59Same cutouts, same edge, different placement, different shape. The other 8 gives nothing new — the two 8s are mirror images. So this cutout has six builds and three different figures: a rhombus, a parallelogram and a kite. Count that properly — up to congruence, not by name. Two figures sharing a name can still be two figures. Now a cutout with three different sides: 6, 9 and 12. Nothing is symmetric any more, so nothing collapses. Six builds, six different quadrilaterals.

6:39Turn the twin round on each edge and you get three parallelograms, with side pairs 6 and 9, then 9 and 12, then 6 and 12. Which is the list of ways to leave one edge out — because the one left out is the one you joined along. Turn the twin over on each edge instead and you get three kites. So the three cutouts have gone one, three, six — and sweeping whole families of each kind, the counts hold every time.

7:11The more symmetric the cutout, the fewer different figures it can make. Symmetry costs you variety. One of those six kites is not shaped like a kite at all. It caves inwards. Here is why, and it is only about the corners at the two ends of the joined edge. Turning the twin over lands a copy of each of those corners beside the original, so each of them doubles.

7:39This cutout's widest corner is 104.5 degrees. Double it and you are past 180, past straight, and the figure has to bend back. That corner faces the 12, so it sits at an end of the 6 and at an end of the 9, and at neither end of the 12. Which is why exactly two of the three cave in, and the one built on the longest edge comes out clean.

8:07Turning the twin round never does this. Half a turn splits those corners between two places instead of doubling them. There is a cutout that sits exactly on the line between those two behaviours. Take one with a square corner — 9, 12 and 15. Turn it over across either edge running into that corner. The doubled angle is not more than straight and not less. It is exactly straight.

8:39So the corner stops being a corner: three of the four points lie in a line. There is no quadrilateral. You have made a bigger triangle. Across twenty-seven builds, every caved-in figure had a corner wider than a right angle at the joined edge, every flat one a square corner there, and the rest came out convex. One measurement at the joined edge tells you which of the three you are about to get.

9:10Last thing. Take the equilateral cutout again and halve it — 4 instead of 8. The rhombus you get has sides of 4 instead of 8, and its diagonal is 4 instead of 8. Every length has halved. And every corner is untouched: still 60, still 120. That is what the size of a cutout controls, and what it does not. Which brings us to the thing worth carrying away.

9:39At no point did we recognise a shape by looking at it. We read lengths off the sides or angles off the corners, and the name followed. The picture tells you where to look. It never tells you the answer.