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?

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Jul 12, 2017

Answer:

#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#

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