Triangles

Find the sides of an equilateral triangle ABC where a trapezium BCPQ is drawn with BP=CQ=2 , PQ=1 and AP+AQ=sqrt7 . Note: there are 2 possible interpretations.

Given that ABCD is a square, M is the mid point of AD and CP is perpendicular to MB with P on MB, prove DP = DC.

A hexagon, with sides alternately a and b units in length, is inscribed in a circle. How big is the radius of the circle?

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

You have pitched your tent (the red triangle) on an island. Can you move it to the position shown by the purple triangle making sure you obey the rules?

The sides of a triangle are 25, 39 and 40 units of length. Find the diameter of the circumscribed circle.

Imagine an infinitely large sheet of square dotty paper on which you can draw triangles of any size you wish (providing each vertex is on a dot). What areas is it/is it not possible to draw?

How would you move the bands on the pegboard to alter these shapes?

A group activity using visualisation of squares and triangles.

What shapes can you make by folding an A4 piece of paper?