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# Coordinates of Corners

Ahmed found some collections of which made squares.

Beth from Oxford found a more complete pattern

In order to be a square it needs two things

If you take the second coordinate away from the first coordinate, the third coordinate away from the second coordinate, the fourth coordinate away from the third coordinate and the first coordinate away from the fourth coordinate, you always get a pattern which looks like

$$\pmatrix{x\cr y},\quad\pmatrix{y\cr-x},\quad\pmatrix{-x\cr -y},\pmatrix{-y\cr x}$$

Why do you think this is? You might want to look at vectors round a square next.

Susie suggested a different way of deciding whether or not the pattern is a square.

If we consider the six distances between the four points and we have four of them of size $a$, for some $a$, and two of them of size $a\times \sqrt{2}$, then we have a square. So we need to compute all the distances between the four points. If they have coordinates $(x_1,y_1), (x_2,y_2), (x_3, y_3), (x_4, y_4)$, then the six distances are $\sqrt{(x_i-x_j)^2 +(y_i-y_j)^2}$ for $1\leq i<j \leq 4$. After that, we arrange them in order and we see whether or not the first four distances are the same, say $a$, and the next two are each of length $a\times \sqrt2$. If they are, then they form a square, otherwise, they don't.

Well done to everyone who sent in a solution. Keep exploring!## You may also like

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Age 11 to 16

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Ahmed found some collections of which made squares.

- (8,3), (7,8), (2,7), (3,2)
- (16,19), (18,22), (21,20), (19,17)
- (4,20), (21,19), (20,2), (3,3)

- $\pmatrix{7\cr 8}-\pmatrix{8\cr 3}=\pmatrix{-1\cr 5}=\pmatrix{2\cr 7}-\pmatrix{3\cr 2}$

Beth from Oxford found a more complete pattern

In order to be a square it needs two things

- All the sides are the same length
- All the angles are right angles

If you take the second coordinate away from the first coordinate, the third coordinate away from the second coordinate, the fourth coordinate away from the third coordinate and the first coordinate away from the fourth coordinate, you always get a pattern which looks like

$$\pmatrix{x\cr y},\quad\pmatrix{y\cr-x},\quad\pmatrix{-x\cr -y},\pmatrix{-y\cr x}$$

Why do you think this is? You might want to look at vectors round a square next.

Susie suggested a different way of deciding whether or not the pattern is a square.

If we consider the six distances between the four points and we have four of them of size $a$, for some $a$, and two of them of size $a\times \sqrt{2}$, then we have a square. So we need to compute all the distances between the four points. If they have coordinates $(x_1,y_1), (x_2,y_2), (x_3, y_3), (x_4, y_4)$, then the six distances are $\sqrt{(x_i-x_j)^2 +(y_i-y_j)^2}$ for $1\leq i<j \leq 4$. After that, we arrange them in order and we see whether or not the first four distances are the same, say $a$, and the next two are each of length $a\times \sqrt2$. If they are, then they form a square, otherwise, they don't.

Well done to everyone who sent in a solution. Keep exploring!

P is a point on the circumference of a circle radius r which rolls, without slipping, inside a circle of radius 2r. What is the locus of P?

Draw a square. A second square of the same size slides around the first always maintaining contact and keeping the same orientation. How far does the dot travel?