Triangles

Prove that a triangle with sides of length 5, 5 and 6 has the same area as a triangle with sides of length 5, 5 and 8. Find other pairs of non-congruent isosceles triangles which have equal areas.

ABCDEFGH is a 3 by 3 by 3 cube. Point P is 1/3 along AB (that is AP : PB = 1 : 2), point Q is 1/3 along GH and point R is 1/3 along ED. What is the area of the triangle PQR?

If you know the sizes of the angles marked with coloured dots in this diagram which angles can you find by calculation?

Weekly Problem 27 - 2013
The diagram shows a parallelogram inside a triangle. What is the value of $x$?

Weekly Problem 29 - 2013
An equilateral triangle is drawn inside a rhombus, both with equal side lengths. What is one of the angles of the rhombus?

Using LOGO, can you construct elegant procedures that will draw this family of 'floor coverings'?

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

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?

Weekly Problem 13 - 2012
The diagram shows contains some equal lengths. Can you work out one of the angles?