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Just Opposite

A and C are the opposite vertices of a square ABCD, and have coordinates (a,b) and (c,d), respectively. What are the coordinates of the vertices B and D? What is the area of the square?

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Ladder and Cube

A 1 metre cube has one face on the ground and one face against a wall. A 4 metre ladder leans against the wall and just touches the cube. How high is the top of the ladder above the ground?

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Beelines

Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?

Pentagon

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

It is often advisable to start with a simple case of the same problem. Here you might like first to think of the same problem for a triangle where you are given the midpoints of the sides and have to find the vertices. The problem suggests doing this with mid-point coordinates $(6, 0)$, $(6.5, 2)$, $(7.5, 1)$. Write down and solve some simultaneous equations and find the vertices.

Then if you choose any set of 3 mid-points could you find the vertices?

You can then extend the method to pentagons.

Then consider the differences between cases of polygons with an odd number of sides and polygons with an even number of sides.

Compare this to the problem Polycircles.

Vassil Vassilev from Lawnswood High School, has an idea for constructing pentagons from the midpoints of edges based on nested pentagons, and 5 pointed stars within them, which are all enlargements of each other. You might like to play with this idea.