Hello, this is problem nine on the grade 11 and 12 2025 Math Kangaroo. Robert wants to have four points so that the distances between each pair of points are different. Which of the points A, B, C, D, and E should he remove? Answer choice A is A, B is point B, C is point C, D is point D, and E is point E. You know, try and solve this one by yourself. My lines should maybe be a useful little hint. Would you have a different approach? Feel free to try that one instead. Okay, so yeah, this is another sort of problem where all the work is going to be done in the diagram. So let's really zoom in close to that diagram. Okay, so we have this diagram here. And I've already sort of taken the liberty of labeling pairs that were the same. And yeah, there's this handy little thing called the Pythagorean theorem, if you haven't heard of it, that sort of tells us if we go down the same amount, like go down and then right, or down and left, but go vertically and horizontally, sort of the same distance, we should be left with two distances that are the same. So if you notice those two greens, or two signs in the graph, or two magentas in the graph, those should be the same length by the Pythagorean theorem. So if we just identify a lot of the pairs that have the same horizontal and verticals, we are able to identify pretty much all the pairs that are of the same length. And then from there, we can sort of go about this one systematically. And we can say that we need to remove, let's go with orange first, we need to remove one of A, B, C, or D, because if we remove E, then we'd be left with two orange pairs. And we don't want that, so we need to remove one of A, B, C, or D. Similarly, we can do the same thing with the blue pairs. So that tells us that A, D, C, or E should be removed. So A, D, C, or E should be removed. And then finally, we have these magenta, and these are actually magenta triplets, which tell us that we actually need two of them to be removed. And the only way you can remove two magenta lines are if we cut off a point that has two magenta lines connected to it. And if we notice, that's actually only D. So if we're to remove anyone, it should be like the one sort of in common over all of these, which is D, D, and D. And if we were to actually remove D, we could ignore all of these. So I'm going to maybe black them out. To where, yeah, we just don't want to pay attention to those. They're all the same color, but they've all been removed. Okay. So yeah, then we're left with only one orange pair, one magenta pair, and then one blue pair. And these are all of different lengths. So if we wanted to remove anything, we should almost definitely remove D.

Math Kangaroo

unique distances

Pythagorean theorem

point removal

graphical analysis