Pythagoras' theorem

problem
Generating triples

Age

14 to 16

Challenge level

Sets of integers like 3, 4, 5 are called Pythagorean Triples, because they could be the lengths of the sides of a right-angled triangle. Can you find any more?
problem
Three cubes

Age

14 to 16

Challenge level

Can you work out the dimensions of the three cubes?
problem
Equilateral areas

Age

14 to 16

Challenge level

ABC and DEF are equilateral triangles of side 3 and 4 respectively. Construct an equilateral triangle whose area is the sum of the area of ABC and DEF.
problem
Grid lockout

Age

14 to 16

Challenge level

What remainders do you get when square numbers are divided by 4?
problem
Zig zag

Age

14 to 16

Challenge level

Four identical right angled triangles are drawn on the sides of a square. Two face out, two face in. Why do the four vertices marked with dots lie on one line?
problem
Rhombus in rectangle

Age

14 to 16

Challenge level

Take any rectangle ABCD such that AB > BC. The point P is on AB and Q is on CD. Show that there is exactly one position of P and Q such that APCQ is a rhombus.
problem
Semi-detached

Age

14 to 16

Challenge level

A square of area 40 square cms is inscribed in a semicircle. Find the area of the square that could be inscribed in a circle of the same radius.
problem
Inscribed in a circle

Age

14 to 16

Challenge level

The area of a square inscribed in a circle with a unit radius is 2. What is the area of these other regular polygons inscribed in a circle with a unit radius?
problem
The spider and the fly

Age

14 to 16

Challenge level

A spider is sitting in the middle of one of the smallest walls in a room and a fly is resting beside the window. What is the shortest distance the spider would have to crawl to catch the fly?
problem
Where to land

Age

14 to 16

Challenge level

Chris is enjoying a swim but needs to get back for lunch. How far along the bank should she land?