Some(?) of the Parts

A circle touches the lines OA, OB and AB where OA and OB are perpendicular. Show that the diameter of the circle is equal to the perimeter of the triangle

Ladder and Cube

A 1 metre cube has one face on the ground and one face against a wall. A 4 metre ladder leans against the wall and just touches the cube. How high is the top of the ladder above the ground?

At a Glance

The area of a regular pentagon looks about twice as a big as the pentangle star drawn within it. Is it?

Equilateral Areas

Age 14 to 16
Challenge Level

Why do this problem

This problem encourages learners to consider familiar mathematical ideas in less less familar contexts. The problem will make connections with triangluar numbers and Pythagoras' theorem. If non-standard units of area (equilateral triangles) are used calculation can be made easier.

Possible approach

Draw the two triangles and ask the group to consider how they would calculate their areas. Share ideas. If the suggestion does not arise from the group - draw one of the triangles on an isometric grid and ask the question again, seeking to draw attention to areas using areas of triangles as the unit of measurement.

Now pose the problem and leave the group to experiment, raise and test conjectures.

Key questions

What about triangles whose sides are not of integer length?

Possible support

Find a rule for the areas of all equilateral triangles with a side of integer length. Can they use this to generate some examples?

Possible extension

Do the findings work when using areas measured in square units?

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