Similarity and congruence

problem

Squareflake

Age

16 to 18

Challenge level

A finite area inside and infinite skin! You can paint the interior of this fractal with a small tin of paint but you could never get enough paint to paint the edge.
problem

Strange Rectangle

Age

16 to 18

Challenge level

ABCD is a rectangle and P, Q, R and S are moveable points on the edges dividing the edges in certain ratios. Strangely PQRS is always a cyclic quadrilateral and you can find the angles.
problem

Flower

Age

16 to 18

Challenge level

Six circles around a central circle make a flower. Watch the flower as you change the radii in this circle packing. Prove that with the given ratios of the radii the petals touch and fit perfectly.
problem

Bus Stop

Age

14 to 16

Challenge level

Two buses leave at the same time from two towns Shipton and Veston on the same long road, travelling towards each other. At each mile along the road are milestones. The buses' speeds are constant and in the ratio 5 to 4. The buses travel to and fro between the towns. What milestones are at Shipton and Veston?
problem

Is a Square a Rectangle?

Age

7 to 11

Challenge level

How many rectangles can you find in this shape? Which ones are differently sized and which are 'similar'?
problem

Number the Sides

Age

7 to 11

Challenge level

The triangles in these sets are similar - can you work out the lengths of the sides which have question marks?
problem

Pinhole Camera

Age

11 to 14

Challenge level

Make your own pinhole camera for safe observation of the sun, and find out how it works.
problem

Triangular Tantaliser

Age

11 to 14

Challenge level

Draw all the possible distinct triangles on a 4 × 4 dotty grid. Convince me that you have all possible triangles.
problem

Guillotine

Age

11 to 14

Challenge level

Weekly Problem 9 - 2006
What fraction of the area of the rectangle is shaded?
problem

Fitting In

Age

14 to 16

Challenge level

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