Corner Cut
Weekly Problem 23 - 2017
Three small equilateral triangles of the same size are cut from the corners of a larger equilateral triangle. What is the side length of the small triangles?
Three small equilateral triangles of the same size are cut from the corners of a larger equilateral triangle. What is the side length of the small triangles?
Age
11 to 14
Challenge level
Problem
Image
Three small equilateral triangles of the same size are cut from the corners of a larger equilateral triangle with sides of length $6\text{cm}$, as shown.
The sum of the perimeters of the small equilateral triangles is the same as the perimeter of the hexagon.
What is the side length of one of the small triangles?
If you liked this problem, here is an NRICH task that challenges you to use similar mathematical ideas.
Student Solutions
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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 $1.5\text{cm}$.