The kite: why one diagonal bisects the other at right angles, and the angles too

0:00Two identical triangles, 6, 9 and 12. Join them along their 12 edges and turn one of them over. The 12 goes inside. You have a quadrilateral with 6 on both upper sides, 9 on both lower ones, and the 12 down the middle. That figure is a kite, and everything it does comes from one thing: it has a line of symmetry, and you built it. Fold it along that 12 and the two halves land exactly on each other.

0:33Now here is the discipline for the rest of this video. We forget how it was made and read everything off the figure itself — off its four lengths and its four corners. A shape does not know how it was built. Only its measurements are available to it. So, the definition, and it needs reading slowly. A kite is a quadrilateral you can letter so two sides that MEET are equal, and the other two, which also meet, are equal as well.

1:06Two pairs. Each pair shares a corner. And the two pairs share no side between them. Here the two 6s meet at the top and the two 9s at the bottom, and those corners are opposite each other. That is not decoration. If the pairs shared a side you would have one pair counted twice. Everything else in this video falls out of that sentence and nothing is added to it.

1:33No angle is mentioned. No parallel is mentioned. Only lengths, and only which lengths sit next to which. There is exactly one word you can get wrong here, and it changes the whole shape. Say opposite instead of neighbouring. Arrange the same four lengths so the equal ones face each other rather than meet. That is a parallelogram — different figure, different properties. The definition is worded the way it is to block that swap.

2:07And one pair alone is not enough either. A figure with a single equal neighbouring pair has none of the properties: no diagonal halving the other, no square crossing. Half the definition buys nothing at all. Before the proof, it is worth being clear about what a kite is not promised. Not parallel sides: sweep twenty-eight kites and not one has a single parallel pair, so a kite need not be a trapezium.

2:39Not equal angles: the two at the sides here match and the other two do not, and no kite in the family has all four the same. Not equal diagonals either. Ours are 12 and about 8.7. The four corners still total a full turn, but that is true of every quadrilateral and is not a kite property. So a kite starts with almost nothing. Which makes what comes next surprising: from lengths alone, and only lengths, you get an angle result and a right angle.

3:16First, which diagonal are we talking about? Because only one of them does anything. Look at the tick marks and nothing else. Single ticks on the two upper sides, double ticks on the two lower ones. The special diagonal joins the corner where the single ticks meet to the corner where the double ticks meet. You need not know which edge the cutouts were joined along. The marks are enough.

3:44And to be sure that is a real reading and not a habit of how we drew it, letter the same kite from a different corner. Now it is the second diagonal that the marks pick out — and it is the same 12 as before, the same segment. The marks follow the shape. The lettering does not. Now cut the kite along that diagonal and look at the two triangles it makes.

4:13The first has a 6, a 9 and the 12. So does the second. Three pairs of matching sides. Three lengths fix a triangle completely, so these two are the same triangle drawn twice. Notice what went into that: the two equal pairs, and the diagonal belonging to both halves. Nothing else. No angle was assumed anywhere. Try it across the OTHER diagonal and it fails at once — one of those triangles has 9, 9 and 8.7, the other 6, 6 and 8.7.

4:51One diagonal gives you a congruence. The other gives you nothing. Read the matching parts off, and the first consequence arrives. The angle the diagonal makes with the upper side on the left equals the one it makes on the right. So the top corner is cut into two equal halves — about 46 and a half each. The same at the other end: the bottom corner splits into two halves of 29.

5:21One congruence, two halved corners, on every kite in both families — twenty-eight, then sixty-eight more. But stop and notice what you have NOT got. Nothing yet about the other diagonal. Nothing yet about a right angle. The first congruence does not reach them. So take a second congruence, and it needs the first one's answer as an input. Look at the two small triangles that meet at the crossing, one on each side of the special diagonal.

5:56Each has a 9 — the equal pair. Each has the piece of the special diagonal down to the crossing — shared. And the angle between those two sides matches in both, ONLY because the first congruence already halved the bottom corner. Two sides and the angle between them: the small triangles are the same triangle. So the two pieces of the other diagonal are equal — the special diagonal cuts it exactly in half.

6:28That is the second result, and it was not available a moment ago. One thing left, and it comes from the same pair of small triangles. Match the angles at the crossing. The one on the left equals the one on the right. But those two sit on a straight line — the two parts of the other diagonal meeting at a point. Two equal angles making 180 between them. Each is 90.

6:57So the diagonals cross at a right angle, proved rather than observed. The order is the content. One congruence for the halved corners; a second, which needs the first, for the halved diagonal and the right angle. Two steps, not one, and they do not commute. Now look at what the other diagonal does, which is nothing. It is not halved by anything. The special diagonal cuts it in half and gets nothing back — its own pieces here are 4.1 and 7.9.

7:35Nor does it halve the corners it passes through. Those two corners do come out equal to each other, but equal and halved are different claims and only the first holds. Across every kite in both families, not one has its second diagonal halving a corner. And that asymmetry is the whole difference between a kite and a rhombus. In a rhombus both diagonals do all of it. In a kite, one does everything and one does nothing.

8:09Here is a consequence people find odd. Give me two diagonals — say 8 and 6 — and ask me to build the kite. The 6 has to be halved by the 8. That much is forced. But where along the 8 does it cross? Nothing has said. Slide the crossing from the middle outwards and every position gives a different kite — nine positions, nine figures. Ask instead that the two diagonals halve EACH OTHER and the crossing has nowhere left to go.

8:428 and 6 then give exactly one figure, the rhombus of side 5 — and it is one of those nine, the one at halfway. So two lengths pin down a rhombus and do not pin down a kite. The extra freedom is exactly the property the kite is missing. Last, where the kite sits among the others. Every rhombus is a kite. Four equal sides give you two separate neighbouring pairs several times over.

9:14Sweep thirty-five figures of both kinds and the ones that are kites AND parallelograms are exactly the seven rhombi. That overlap has a name, and the name is rhombus. Can a kite be a rectangle? Yes, but only the square — both neighbouring pairs and four square corners. One right angle is not four. And a warning about pictures. A kite can cave inwards. When it does it keeps every property — the diagonal still halves the other, still squarely — but the crossing sits outside it.

9:52Which is the discipline we started with. The properties are statements about lengths and angles, not about where the ink happens to land.