A square is comprised of two 45-45-90 right triangles. The hypotenuse of a 45-45-90 right triangle follows the rule below, where  is the length of the sides. 

In this instance, is equal to 6.

To find the diagonal of the square, we effectively cut the square into two triangles.

The pattern for the sides of a is .

Since two sides are equal to , this triangle will have sides of .

Therefore, the diagonal (the hypotenuse) will have a length of .

To find the diagonal of a square, we must use the side length to create a 90 degree triangle with side lengths of , , and a hypotenuse which is equal to the diagonal.

Pythagorean’s Theorem states , where a and b are the legs and c is the hypotenuse.

Take  and  and plug them into the equation for  and :

After squaring the numbers, add them together:

Once you have the sum, take the square root of both sides:

Simplify to find the answer: , or .

To find the diagonal of a square we must use the side lengths to create a 90 degree triangle with side lengths of 7 and a hypotenuse which is equal to the diagonal.

We can use the Pythagorean Theorem here to solve for the hypotenuse of a right triangle.

The Pythagorean Theorem states , where a and b are the sidelengths and c is the hypotenuse.

Plug the side lengths into the equation as  and :

Square the numbers:

Add the terms on the left side of the equation together:

Take the square root of both sides:

Therefore the length of the diagonal is 9.9.

Perimeter = side * 4

48 = side * 4

Side = 12

We can break up the square into two equal right triangles. The diagonal of the sqaure is then the hypotenuse of these two triangles.

Therefore, we can use the Pythagorean Theorem to solve for the diagonal:

Assign variables such that

One side of ABCD = a

and One side of EFGH = e

Note that all sides are the same in a square. Since the perimeter is the sum of all sides, according to the question:

4a = 3 x 4e = 12e or a = 3e

From that area of EFGH is 25,

e x e = 25 so e = 5

Substitute a = 3e so a = 15

We aren’t done. Since we were asked for the area of ABCD, this is a x a = 225.

Let S be the original side length. S*S would represent the original area. Doubling the side length would give you 2S*2S, simplifying to 4*(S*S), giving a new area of 4x the original, or 144.

Area of a square in terms of each of its sides:

  Area = S x S

Perimeter of a square:

  Perimeter = 4S

So if 'the perimeter of a square is equal to twice its area':

  2 x Area = Perimeter

  2 x [S x S] = [4S]; divide by 2:

  S x S = 2S; divide by S:

  S = 2

If Freddie uses x feet of fencing makes a square, each side must be x/4 feet long. The area of this square is (x/4)² = x²/16 = p square feet.

If Freddie uses one third of x feet of fencing makes a square, each side must be x/12 feet long. The area of this square is (x/12)² = x²/144 = 1/9(x²/16) = 1/9(p) = p/9 square feet.

Alternate method:

The scale factor between the small perimeter and the larger perimeter = 1 : 3. Since we're comparing area, a two-dimensional measurement, we can square the scale factor and see that the ratio of the areas is 1² : 3² = 1 : 9.