Circles on a triangle
Find the missing length on this diagram.
Age
14 to 16
Challenge level
Problem
On this diagram, the circles centred at A and C touch.
Find the length CE.
Image
This problem is taken from the World Mathematics Championships
Student Solutions
The radii of the circles along the side of the triangle are also shown in the diagram below.
Image
Using the similar triangles in the diagram
Triangles AEB and CED are similar. The scale factor from CED to AEB is the same as the scale factor from CD to AB, ie from 4 cm to 6 cm.
This is the same as adding half of the original length.
Image
So to find the length AE from the length CE, you would need to add half of the length CE.
But the length you need to add is AC, which is 10 cm.
So CE must be 20 cm.
Drawing a line to make another similar triangle
6 cm can be split into 4 cm and 2 cm.
Image
The triangle with sides 2 cm and 10 cm is similar to the triangle with sides 4 cm and ?.
So the ratios 2 : 10 and 4 : ? are the same.
So ? = 20 cm.