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?

Nicely Similar

Stage: 4 Challenge Level:

Do you know this diagram?

diagram with similar right-angled triangles

A right-angled triangle has an extra line descending from the original right angle, perpendicular to the hypotenuse.

If the hypotenuse (base) length is 100cm and if that extra line splits the base into 36cm and 64cm parts, what were the side lengths for the original right-angled triangle?

Sometimes we use the word "scale" to mean enlarge or scale up - for another way to see this arrangement of right-angled triangles you may like to look at the problem called a Matter of Scale .

Generalising. Pythagoras' theorem. Polynomials. smartphone. Manipulating algebraic expressions/formulae. Factorisation (algebraic). Similarity. Simultaneous equations. Creating expressions/formulae. Mathematical reasoning & proof.