Set Square

A triangle PQR, right angled at P, slides on a horizontal floor with Q and R in contact with perpendicular walls. What is the locus of P?

Bendy Quad

Four rods are hinged at their ends to form a convex quadrilateral. Investigate the different shapes that the quadrilateral can take. Be patient this problem may be slow to load.

Biggest Bendy

Four rods are hinged at their ends to form a quadrilateral with fixed side lengths. Show that the quadrilateral has a maximum area when it is cyclic.

Quads

Stage: 4 and 5 Challenge Level:

Why do this problem?

It involves learners in visualisation and working with dynamic images and in conjecturing and proving. They have to select the mathematical information, methods and tools to use.

Possible approach

You might use this problem in connection with work on circle theorems or later as an application and revision of previous work. You might accept the statement that a line joining the centres of two circles is perpendicular to the common chord or you might ask learners to prove this.

Key questions

What stays invariant as the diagram changes?

What is changing?

What do you notice about the quadrilateral PQRS as the diagram changes?

What other questions can we ask about this figure?

Possible extension

What happens if one of the vertices A, B, C or D moves on top of another one or on top of the
point X (degenerate cases)?

What happens if ABCD is not convex?

Try the problems: Cyclic Quads and Chords

Possible support

Draw the dynamic diagram for yourself. Geogebra is very easy to use and it is free.

Manipulating algebraic expressions/formulae. Inequality/inequalities. Patterned numbers. Parallel lines. Number theory. Perpendicular lines. Dynamic geometry. Quadrilaterals. Mathematical reasoning & proof. Circle theorems.

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