Question

A square and an equilateral triangle have the same perimeter. If the diagonal of the square is `12sqrt2` cm, what is the area of the triangle?

Answer 1

`64 sqrt(3) " cm"^2`

Explanation:

Let `A_s` be the area of the square, `P_s` be the perimeter of the square and `a_s` be the length of a side of the square. (All sides have equal lengths.)

Let `A_t` be the area of the triangle, `P_t` be the perimeter of the triangle and `a_t` be the length of a side of the triangle. (All sides have equal lengths.)

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1) As we know the length of the diagonal of the square, we can compute the length of a side of the square using the Pythagoras formula:

`a_s^2 + a_s^2 = d^2`

`=> 2 a_s^2 = (12 sqrt(2) )^2`

`=> a_s^2 = 12^2`

`=> a_s = 12 "cm"`

2) Knowing the length of one side of the square (and thus knowing all lengths of a square), we can easily compute the square's perimeter:

`P_s = 12 * 4 = 48 "cm"`

3) We know that the square and the equilateral triangle have the same perimeter, thus

`P_t = 48 "cm"`

4) As all sides have the same length in an equilateral triangle, the length of one side is

`a_t = P_t / 3 = "16 cm"`

5) Now, to compute the area of the equilateral triangle, we need the height `h` which can be computed with the Pythagoras formula again:

`h^2 + (a_t/2)^2 = a_t^2`

`=> h^2 + 8^2 = 16^2`

`=> h^2 = 192 = 64 * 3`

`=> h = 8 sqrt(3) "cm"`

6) At last, we can compute the area of the triangle:

`A_t = 1/2 * h * a_t = 1/2 * 8 sqrt(3) * 16 = 64 sqrt(3) " cm"^2`