A square and an equilateral triangle have equal perimeters. The area of the triangle is 2\sqrt {3} square inches. What is the number of inches in the length of the diagonal of the square?
Please give me a full clear explanation with the answer. Thanks!!!!!!!
the triangle of side s has altitude s√3/2
So, it has area 1/2 (s)(s√3/2) = s^2√3/4
So, if s^2√3/4 = 2√3,
s = √8
So, the triangle has perimeter 3√8
If the square has perimeter 3√8, it has side 3√8/4 = 3/2 √2
The diagonal of a square has length s√2, so in this case, the diagonal is 3.
Let the side of the equilateral triangle be x
area of equilateral triangle = (1/2)x^2 sin60°
= (1/2)x^2 (√3/2) = (√3/4)x^2
(√3/2)x^2 = 2/√3
3x^2 = 4
x = 2/√3
perimeter of triangle = 3x = 6/√3
which is equal to the perimeter of the square, so each side of the square is 6/(4√3) = 3/(2√3)
let the diagonal be d
d^2 = (3/(2√3) )^2 + (3/(2√3) )^2
= 9/12 +9/12 = 18/12 = 6/4
d = √6/√4 = √6/2
made a silly error ...
(√3/2)x^2 = 2/√3
3x^2 = 4
x^2 = 8/3
x = √8/√3
I also read your area of the triangle as 2/√3
whereas Steve took it as 2√3
That's what happens when TEX gets mixed in with text!
no its 3
To solve this problem, we need to find the relationship between the side lengths of the square and the equilateral triangle.
Let's assume the side length of the square is "s" inches.
The perimeter of a square is given by the formula: P = 4s (since all sides of a square are equal).
The perimeter of an equilateral triangle is given by the formula: P = 3s (since all sides of an equilateral triangle are equal).
Given that the square and the equilateral triangle have equal perimeters, we can set up an equation:
4s = 3s
If we subtract 3s from both sides, we get:
s = 0
Wait a minute! This doesn't seem right. We cannot have a side length of zero inches for the square.
Let's check if our calculations are correct by using the area of the triangle given in the problem.
The formula for the area of an equilateral triangle is given by:
A = (sqrt(3) / 4) * s^2, where s is the side length of the triangle.
Given that the area of the triangle is 2√3 square inches, we can set up an equation:
2√3 = (sqrt(3) / 4) * s^2
To simplify, we can multiply both sides by 4:
8√3 = sqrt(3) * s^2
Now, let's square both sides to get rid of the square root:
64 * 3 = 3 * s^4
Dividing both sides by 3:
64 = s^4
Taking the fourth root of both sides:
s = 2
Now we have determined that the side length of the square is 2 inches.
Since the square is a regular quadrilateral, we know that the length of the diagonal (d) can be found using the Pythagorean theorem:
d^2 = s^2 + s^2
d^2 = 2^2 + 2^2
d^2 = 4 + 4
d^2 = 8
Taking the square root of both sides to solve for d:
d = √8
Simplifying the square root:
d = √(4 * 2)
d = √4 * √2
d = 2 * √2
Therefore, the number of inches in the length of the diagonal of the square is 2√2 inches.