### Fitting In

The largest square which fits into a circle is ABCD and EFGH is a square with G and H on the line CD and E and F on the circumference of the circle. Show that AB = 5EF. Similarly the largest equilateral triangle which fits into a circle is LMN and PQR is an equilateral triangle with P and Q on the line LM and R on the circumference of the circle. Show that LM = 3PQ

### Look Before You Leap

Can you spot a cunning way to work out the missing length?

### Triangle Midpoints

You are only given the three midpoints of the sides of a triangle. How can you construct the original triangle?

# Triangle in a Trapezium

##### Age 11 to 16 Challenge Level:

For each of the starting points, it might be a good idea to label some of the side lengths and heights with letters.

Starting Point 1

How would you work out the area of the two unshaded triangles? You might need to choose some letters to represent certain quantities...
Using the same letters, can you work out the area of the trapezium?

Starting Point 2

How does triangle BEG compare with triangle AEF? How do you know?
How do we know that triangle CED and triangle CFD have the same area?

Starting Point 3

The trapezium has been rotated $180^{\circ}$ around F.
Can you prove that ABGH is a parallelogram? It might help to mark angles that you know are equal, and angles that you know add up to $180^{\circ}$.