Dynamic geometry

article

Using GeoGebra

Never used GeoGebra before? This article for complete beginners will help you to get started with this free dynamic geometry software.

problem

Isosceles

Age

11 to 14

Challenge level

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.

problem

A roll of patterned paper

Age

14 to 16

Challenge level

A design is repeated endlessly along a line - rather like a stream of paper coming off a roll. Make a strip that matches itself after rotation, or after reflection

problem

Two shapes and printer ink

Age

14 to 16

Challenge level

If I print this page which shape will require the more yellow ink?

problem

Trapezium four

Age

14 to 16

Challenge level

The diagonals of a trapezium divide it into four parts. Can you create a trapezium where three of those parts are equal in area?

Trapezium made of wooden tangram pieces, including a square and a parallelogram.

problem

Quad in quad

Age

14 to 18

Challenge level

Join the midpoints of a quadrilateral to get a new quadrilateral. What is special about it?

problem

Arrowhead

Age

14 to 16

Challenge level

The points P, Q, R and S are the midpoints of the edges of a non-convex quadrilateral.What do you notice about the quadrilateral PQRS and its area?

problem

The medieval octagon

Age

14 to 16

Challenge level

Medieval stonemasons used a method to construct octagons using ruler and compasses... Is the octagon regular? Proof please.

problem

Three balls

Age

14 to 16

Challenge level

A circle has centre O and angle POR = angle QOR. Construct tangents at P and Q meeting at T. Draw a circle with diameter OT. Do P and Q lie inside, or on, or outside this circle?

problem

Move the ends of the lines at points B and D around the circle and find the relationship between the length of the line segments PA, PB, PC, and PD. The length of each of the line segments is recorded next to the diagram.

problem

One reflection implies another

Age

14 to 16

Challenge level

When a strip has vertical symmetry there always seems to be a second place where a mirror line could go. Perhaps you can find a design that has only one mirror line across it. Or, if you thought that was impossible, could you explain why ?

problem

Points in pairs

Age

14 to 16

Challenge level

Move the point P to see how P' moves. Then use your insights to calculate a missing length.

problem

Polycircles

Age

14 to 16

Challenge level

Show that for any triangle it is always possible to construct 3 touching circles with centres at the vertices. Is it possible to construct touching circles centred at the vertices of any polygon?

problem

Bendy quad

Age

14 to 16

Challenge level

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.

problem

Triangle incircle iteration

Age

14 to 16

Challenge level

Keep constructing triangles in the incircle of the previous triangle. What happens?

problem

Napoleon's theorem

Age

14 to 18

Challenge level

Triangle ABC has equilateral triangles drawn on its edges. Points P, Q and R are the centres of the equilateral triangles. What can you prove about the triangle PQR?

problem

Rotations are not single round here

Age

14 to 16

Challenge level

I noticed this about streamers that have rotation symmetry : if there was one centre of rotation there always seems to be a second centre that also worked. Can you find a design that has only one centre of rotation ? Or if you thought that was impossible, could you say why ?

problem

The rescaled map

Age

14 to 16

Challenge level

We use statistics to give ourselves an informed view on a subject of interest. This problem explores how to scale countries on a map to represent characteristics other than land area.

problem

Fixing it

Age

16 to 18

Challenge level

A and B are two fixed points on a circle and RS is a variable diamater. What is the locus of the intersection P of AR and BS?

problem

Cushion ball

Age

16 to 18

Challenge level

The shortest path between any two points on a snooker table is the straight line between them but what if the ball must bounce off one wall, or 2 walls, or 3 walls?

problem

Napoleon's hat

Age

16 to 18

Challenge level

Three equilateral triangles ABC, AYX and XZB are drawn with the point X a moveable point on AB. The points P, Q and R are the centres of the three triangles. What can you say about triangle PQR?

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Dynamic geometry

Using GeoGebra

Isosceles

A roll of patterned paper

Two shapes and printer ink

Trapezium four

Quad in quad

Arrowhead

The medieval octagon

Three balls

Secants interactivity

One reflection implies another

Points in pairs

Polycircles

Bendy quad

Triangle incircle iteration

Napoleon's theorem

Rotations are not single round here

The rescaled map

Fixing it

Cushion ball

Napoleon's hat