can someone help me with hw?

Question

cuz i need help!
If the diagonal of a square has a length of 4[radical]2, then the perimeter of the square is
(1) 8
(2) 16
(3) 16[rad]2
(4) 32

can someone explain it to me and break it down please?

Answer 1

suppose a square has side of length x and diagonal d. then by Pythagorean Theorem, d² = x² + x², d² = 2x², d = √(2x²) = x√2. so if the diagonal is 4√2, the side is 4 and the perimeter is 4(4) = 16

Answer 2

The diagonal divides the square into two right triangles, with the diagonal as the hypotenuse.

So, work on the triangle. 4rad2 is the hypotenuse, and let's call the sides "x". They're both the same, after all; it's coming from a square!

So, the Pythagorean Theorem:

a^2 + b^2 = c^2

Where a and b are the sides, and c is the hypotenuse. Since a and b are the same (x), you can use x for both. And c is 4rad2:

x^2 + x^2 = (4rad2)^2
(4rad2)^2 is 4 * 4 * rad2 * rad2, which is 16 * 2 = 32:
2(x^2) = 32 <---div both sides by 2:
x^2 = 16 <---square root of both sides:
x = 4

This means one SIDE of the square is 4. Four sides in a square. 4 * 4 = 16.

(2)

Answer 3

Drawing the diagonal will create two equal 45-45-90 triangles.

The relationship between a leg and a hypotenuse of a 45-45-90 triangle is that if the leg equals x, the hypotenuse will equal x times root 2.

so if the diagonal equals 4 root 2, the side of the square is 4. then the perimeter is 4 * 4 = 16.

Answer 4

It really helps to draw out the picture. Label each side of the square as x (since they are the same) and draw the diagonal (4 radical 2). If you look at the square you will notice that it is divided into two right triangles. You can now use the Pythagorean Theorem and make the equation x^2 + x^2 equals 4 radical 2 squared. You can probably solve it from there. Hope that helped!