Similarity and congruence

problem
Pent

Age

14 to 18

Challenge level

The diagram shows a regular pentagon with sides of unit length. Find all the angles in the diagram. Prove that the quadrilateral shown in red is a rhombus.
problem
Hex

Age

11 to 14

Challenge level

Explain how the thirteen pieces making up the regular hexagon shown in the diagram can be re-assembled to form three smaller regular hexagons congruent to each other.
problem
Folding squares

Age

14 to 16

Challenge level

The diagonal of a square intersects the line joining one of the unused corners to the midpoint of the opposite side. What do you notice about the line segments produced?
problem
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.
problem
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.
problem
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?
problem
How big?

Age

11 to 14

Challenge level

If the sides of the triangle in the diagram are 3, 4 and 5, what is the area of the shaded square?
problem
From all corners

Age

14 to 16

Challenge level

Straight lines are drawn from each corner of a square to the mid points of the opposite sides. Express the area of the octagon that is formed at the centre as a fraction of the area of the square.
problem
A sameness surely

Age

14 to 16

Challenge level

Triangle ABC has a right angle at C. ACRS and CBPQ are squares. ST and PU are perpendicular to AB produced. Show that ST + PU = AB
problem
Two ladders

Age

14 to 16

Challenge level

Two ladders are propped up against facing walls. At what height do the ladders cross?