Corner Cut

Age 11 to 14

ShortChallenge Level Yellow star

Suppose the small equilateral triangles have sides of length

$x\text{cm}$ . Then, each of these has perimeter

$3x$ , and there are three of these, so the total perimeter is

$3 \times 3x = 9x$ .

The hexagon has three sides where the triangles have been cut from, each of length

$x$ . The other sides are the sides of the larger triangle, less two sides of the smaller triangles. They, therefore, have length

$6-2x$ . There are three of each of these, so the perimeter of the hexagon is

$3x + 3(6-2x)$ .

Since the hexagon has a perimeter equal to the sum of the three small triangles, we get:

$9x = 3x + 3(6-2x)$

Then, expanding the bracket gives:

$9x = 3x + 18 - 6x$

Collecting like terms:

$9x = 18 - 3x$

Adding

$3x$ to each side:

$12x = 18$

Dividing by

$6$ :

$x = 1.5$

Therefore, the small equilateral triangles have sides of length

$1.5\text{cm}$ .

This problem is taken from the UKMT Mathematical Challenges.

You can find more short problems, arranged by curriculum topic, in our short problems collection.

You may also like