Triangles

problem
Kissing triangles

Age

11 to 14

Challenge level

Determine the total shaded area of the 'kissing triangles'.
problem
Three way split

Age

14 to 16

Challenge level

Take any point P inside an equilateral triangle. Draw PA, PB and PC from P perpendicular to the sides of the triangle where A, B and C are points on the sides. Prove that PA + PB + PC is a constant.
problem
Triangle midpoints

Age

14 to 16

Challenge level

You are only given the three midpoints of the sides of a triangle. How can you construct the original triangle?
problem
Fitting in

Age

14 to 16

Challenge level

The largest square which fits into a circle is ABCD and EFGH is a square with G and H on the line CD and E and F on the circumference of the circle. Show that AB = 5EF. Similarly the largest equilateral triangle which fits into a circle is LMN and PQR is an equilateral triangle with P and Q on the line LM and R on the circumference of the circle. Show that LM = 3PQ
problem
Area i'n it

Age

16 to 18

Challenge level

Triangle ABC has altitudes h1, h2 and h3. The radius of the inscribed circle is r, while the radii of the escribed circles are r1, r2 and r3 respectively. Prove: 1/r = 1/h1 + 1/h2 + 1/h3 = 1/r1 + 1/r2 + 1/r3 .
problem
Chain of changes

Age

5 to 7

Challenge level

Arrange the shapes in a line so that you change either colour or shape in the next piece along. Can you find several ways to start with a blue triangle and end with a red circle?
problem
Part the polygons

Age

7 to 11

Challenge level

Draw three straight lines to separate these shapes into four groups - each group must contain one of each shape.
problem
Colouring triangles

Age

5 to 7

Challenge level

Explore ways of colouring this set of triangles. Can you make symmetrical patterns?
problem
Three fingers and a loop of string

Age

5 to 7

Challenge level

Using a loop of string stretched around three of your fingers, what different triangles can you make? Draw them and sort them into groups.
problem
Seven sticks

Age

5 to 7

Challenge level

Explore the triangles that can be made with seven sticks of the same length.