The points A and B are opposite corners of a rectangle. Q1 Copy this diagram. The points A and B are opposite corners of a rectangle. Draw the whole rectangle. B 3 2 1 A 1 2 3

Did you draw this rectangle? Boring! B 3 2 1 A 1 2 3

There are lots of other rectangles with opposite corners at A and B. Can you find any? B 3 2 1 A 1 2 3

Here are four examples: B 3 2 1 A 1 2 3

Example 1: B 3 2 1 A 1 2 3

Example 2: B 3 2 1 A 1 2 3

Example 3: B 3 2 1 A 1 2 3

Example 4: B 3 2 1 A 1 2 3

What do you notice about all of the examples? Have a think about the general pattern. B 3 2 1 A 1 2 3

All of the corners are on the same circle! B 3 2 1 A 1 2 3

This is the circle with centre the mid-point of A and B, and radius so it goes through A and B. 3 2 1 A 1 2 3

Now try and find some rectangles with opposite corners at C and D. Q2 (a) Now try and find some rectangles with opposite corners at C and D. 3 2 1 D C 1 2 3

Now try and find some rectangles with opposite corners at E and F. Q2 (b) Now try and find some rectangles with opposite corners at E and F. E 3 2 1 F 1 2 3

Look again at opposite corners A and B. Q3 (a) Look again at opposite corners A and B. Can you use geometry to prove that the other corners must be on that circle? B 3 2 1 A 1 2 3

Look again at opposite corners A and B. Q3 (b) (harder) Look again at opposite corners A and B. Use algebra to prove the other corners must be on that circle B 3 2 1 A 1 2 3

Q3 (b) Hint 1: Call the top left corner point C, and give it general co-ordinates (x,y) B 3 C (x, y) 2 1 A 1 2 3

Hint 2: Work out the gradient of the line from A to C. Q3 (b) Hint 2: Work out the gradient of the line from A to C. Work out the gradient of the line from C to B. B 3 C (x, y) 2 1 A 1 2 3

Hint 3: The line AC must be perpendicular to the line CB. Q3 (b) Hint 3: The line AC must be perpendicular to the line CB. What does that mean about their gradients? B 3 C (x, y) 2 1 A 1 2 3

Hint 4: The formula for a circle with centre (a,b) and radius r is Q3 (b) Hint 4: The formula for a circle with centre (a,b) and radius r is (x-a)2+(y-b)2=r2 B 3 C (x, y) 2 1 A 1 2 3

Can you generalise your proof? Q4 (harder) Can you generalise your proof? Show that given two diagonal corners the other two corners are always on a circle. B 3 2 1 A 1 2 3

These rectangles all have different areas. Q5 (harder) These rectangles all have different areas. Is there a formula for the area? What is the max area? B 3 2 1 A 1 2 3