1. 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

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

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

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

5. Example 1: B 3 2 1 A 1 2 3

6. Example 2: B 3 2 1 A 1 2 3

7. Example 3: B 3 2 1 A 1 2 3

8. Example 4: B 3 2 1 A 1 2 3

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

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

11. 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

12. 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

13. 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

14. 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

15. 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

16. 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

17. 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

18. 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

19. 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

20. 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

21. 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