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Darts and Kites

Age 14 to 16 Challenge Level:

Explore the geometry of these dart and kite shapes!

Pent

Age 14 to 18 Challenge Level:

The diagram shows a regular pentagon with sides of unit length. Find all the angles in the diagram. Prove that the quadrilateral shown in red is a rhombus.

Pareq Calc

Age 14 to 16 Challenge Level:

Triangle ABC is an equilateral triangle with three parallel lines going through the vertices. Calculate the length of the sides of the triangle if the perpendicular distances between the parallel. . . .

Partly Circles

Age 14 to 16 Challenge Level:

What is the same and what is different about these circle questions? What connections can you make?

A Third of the Area

Age 14 to 16 Short Challenge Level:

The area of the small square is $\frac13$ of the area of the large square. What is $\frac xy$?

Xtra

Age 14 to 18 Challenge Level:

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.

Square Mean

Age 14 to 16 Challenge Level:

Is the mean of the squares of two numbers greater than, or less than, the square of their means?

Pentakite

Age 14 to 18 Challenge Level:

ABCDE is a regular pentagon of side length one unit. BC produced meets ED produced at F. Show that triangle CDF is congruent to triangle EDB. Find the length of BE.

Golden Thoughts

Age 14 to 16 Challenge Level:

Rectangle PQRS has X and Y on the edges. Triangles PQY, YRX and XSP have equal areas. Prove X and Y divide the sides of PQRS in the golden ratio.

Interactive Number Patterns

Age 14 to 16 Challenge Level:

How good are you at finding the formula for a number pattern ?

Age 11 to 14 Challenge Level:

Surprising numerical patterns can be explained using algebra and diagrams...

How Old Am I?

Age 14 to 16 Challenge Level:

In 15 years' time my age will be the square of my age 15 years ago. Can you work out my age, and when I had other special birthdays?

Always Two

Age 14 to 18 Challenge Level:

Find all the triples of numbers a, b, c such that each one of them plus the product of the other two is always 2.

Two Cubes

Age 14 to 16 Challenge Level:

Two cubes, each with integral side lengths, have a combined volume equal to the total of the lengths of their edges. How big are the cubes? [If you find a result by 'trial and error' you'll need to. . . .