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

Two ladders are propped up against facing walls. The end of the first ladder is 10 metres above the foot of the first wall. The end of the second ladder is 5 metres above the foot of the second wall. At what height do the ladders cross?

A circle of radius r touches two sides of a right angled triangle, sides x and y, and has its centre on the hypotenuse. Can you prove the formula linking x, y and r?

A napkin is folded so that a corner coincides with the midpoint of an opposite edge . Investigate the three triangles formed .

It is impossible to trisect an angle using only ruler and compasses but it can be done using a carpenter's square.

Using a ruler, pencil and compasses only, it is possible to construct a square inside any triangle so that all four vertices touch the sides of the triangle.

The diagonals of a trapezium divide it into four parts. Can you create a trapezium where three of those parts are equal in area?

If the hypotenuse (base) length is 100cm and if an extra line splits the base into 36cm and 64cm parts, what were the side lengths for the original right-angled triangle?

What is the same and what is different about these circle questions? What connections can you make?

Explore when it is possible to construct a circle which just touches all four sides of a quadrilateral.

Draw some quadrilaterals on a 9-point circle and work out the angles. Is there a theorem?

Can you make sense of the three methods to work out the area of the kite in the square?

Construct two equilateral triangles on a straight line. There are two lengths that look the same - can you prove it?

Just from the diagram, can you work out the radius of the smaller circles?

Can you find the area of this square inside a circle?

Two semicircles overlap, can you find this length?

Can you find the ratio of the area shaded in this regular octagon to the unshaded area?

Can you find the area of the triangle from its height and two sides?

In the diagram, the radius of the circle is equal to the length AB. Can you find the size of angle ACB?

Can you find the radius of the circle inscribed inside a '3-4-5 triangle'?

What is the largest number of intersection points that a triangle and a quadrilateral can have?

Prove that these two lengths are equal.

A semicircle is drawn inside a right-angled triangle. Find the distance marked on the diagram.

This square piece of paper has been folded and creased. Where does the crease meet the side AD?

Find the missing distance in this diagram with two isosceles triangles