Angles, Polygons and Geometrical Proof (age 14-16)

This is part of our Secondary Curriculum collection of favourite rich tasks arranged by topic.

Cyclic Quadrilaterals

Age 11 to 16
Challenge Level

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

Triangle Midpoints

Age 14 to 16
Challenge Level

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

Making Sixty

Age 14 to 16
Challenge Level

Why does this fold create an angle of sixty degrees?

Circles in Quadrilaterals

Age 14 to 16
Challenge Level

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

Nicely Similar

Age 14 to 16
Challenge Level

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?

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?

Sitting Pretty

Age 14 to 16
Challenge Level

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?

Kite in a Square

Age 14 to 18
Challenge Level

Can you make sense of the three methods to work out what fraction of the total area is shaded?

Trapezium Four

Age 14 to 16
Challenge Level

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

Partly Circles

Age 14 to 16
Challenge Level

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

Napkin

Age 14 to 16
Challenge Level

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

Angle Trisection

Age 14 to 16
Challenge Level

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

Squirty

Age 14 to 16
Challenge Level

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.