A square and a equilateral triangle have the same perimeter.

Question

The sides of the square are 3 times a number, k. The sides of the triangle are one more than twice the length of k. What are the dimensions of the figures?

Pleae answer as soon as possible!

Answer 1

The prompt assumes that the perimeter of a square is equal to the perimeter of an equilateral triangle. Because of this assumption, we can set each of the perimeters equal to each other in terms of k. In the case of the square, each side would be equal to 3k and since there are 4 sides, the perimeter would be equal to 12k. In the case of the triangle, each side would be equal to 2k + 1 and since there are 3 sides, the perimeter would be 6k + 3. These two perimeter values can be set equal to each other to algebraically determine what k is equal to.

12k = 6k + 3

6k = 3

k = 3/6

k = 1/2

Because k is found to be 1/2, we can plug this back into the original equations for side lengths and perimeters.

Side length of square:

3k

= 3(1/2)

= 3/2

Side length of triangle:

2k + 1

= 2(1/2) +1

= 1 + 1

= 2

Perimeter of square:

12k

= 12(1/2)

= 6

Perimeter of triangle:

6k + 3

= 6(1/2) + 3

= 3 + 3

= 6

Note that the perimeter of the square and triangle are both the same. This brings us back to the prompt where it is stated that they are equal. I just showed both methods of solving for the perimeter of both shapes.

The final answer is that the side length of the square is 3/2, the side length of the equilateral triangle is 2, and the perimeter of both the square and triangle is 6.