Similarity and congruence

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

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.

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

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

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

Can you spot a cunning way to work out the missing length?

Find the area of the shaded region created by the two overlapping triangles in terms of a and b?

Triangle ABC is equilateral. D, the midpoint of BC, is the centre of the semi-circle whose radius is R which touches AB and AC, as well as a smaller circle with radius r which also touches AB and AC. What is the value of r/R?