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?

Which has the greatest area, a circle or a square inscribed in an isosceles, right angle triangle?

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.

A hexagon, with sides alternately a and b units in length, is inscribed in a circle. How big is the radius of the circle?

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.

How many different triangles can you make which consist of the centre point and two of the points on the edge? Can you work out each of their angles?

What is the relationship between the angle at the centre and the angles at the circumference, for angles which stand on the same arc? Can you prove it?

What's special about the area of quadrilaterals drawn in a square?

Can you make a right-angled triangle on this peg-board by joining up three points round the edge?

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.

What can you say about the angles on opposite vertices of any cyclic quadrilateral? Working on the building blocks will give you insights that may help you to explain what is special about them.

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

A collection of short Stage 4 problems on geometrical reasoning.