Question

a square and an equilateral triangle have equal perimeters. the area of the triangle is $16\sqrt{3}$ square centimeters. how long, in centimeters, is a diagonal of the square? express your answer in simplest radical form.

Answer 1

To find the length of the diagonal of the square, we can set up equations using the equal perimeters of the square and the equilateral triangle. By solving these equations and applying the Pythagorean theorem, we can find that the length of the diagonal of the square is 6√2 centimeters.

Step-by-step explanation:

To find the length of the diagonal of the square, we need to first find the length of the side of the square. Since the square and the equilateral triangle have equal perimeters, we can set up the equations:

4s = 3a

where s is the side length of the square and a is the side length of the equilateral triangle. We know that the area of the equilateral triangle is given as 16√3, so we can find the side length:

a² = (4/√3) * 16√3

a² = 64 * (√3/√3)

a² = 64

Therefore, a = 8.

Now we can substitute a = 8 back into the equation to find the side length of the square:

4s = 3(8)

4s = 24

s = 6

Finally, we can use the Pythagorean theorem to find the length of the diagonal d:

d² = s² + s² = 36 + 36 = 72

So, d = √72 = 6√2 centimeters.

Answer 2

The diagonal of the square is 6√2 centimeters.

How the diagonal of the square is determined

Let s be the side length of the square, and a be the side length of the equilateral triangle.

For the square, the perimeter is given by

P1 = 4s.

For the equilateral triangle, the perimeter is given by P2 = 3a.

Since the perimeters are equal, we have 4s = 3a

The area of an equilateral triangle is given by the formula A = √3/4 *a²

Given that A = 16√3

√3/4x a² = 16√3

a²/4 = 16

a² = 64

a = √64

a = 8

But 4s = 3a

4s = 3*8

s = 24/4 = 6

For square, the diagonal d is related to the side length (s) by d = s√2.

= 6√2 cm

Therefore, the diagonal of the square is 6√2 centimeters.