The diagonal line cuts the square into two equal triangles.  Their hypotenuse is the diagonal of the square, so we can solve for the hypotenuse.

We need to use the Pythagorean Theorem: \(\displaystyle c^2=a^2+b^2\), where a and b are the legs and c is the hypotenuse.

The two legs have lengths of 8.  Plug this in and solve for c:

\(\displaystyle c^2=8^2+8^2=64+64=128\)

\(\displaystyle c=\sqrt{128}=\sqrt{2\cdot 64}=8\sqrt2\)

You can do this problem in two different ways that lead to the final answer:

1. Pythagorean Theorem

2. Special Triangles (45-45-90)

1. For the first idea, use the Pythagorean Theorem: \(\displaystyle a^2 +b^2 =c^2\), where a and b are the side lengths of the square and c is the length of the diagonal.

\(\displaystyle 4^2+4^2=c^2\)

\(\displaystyle c^2=32\)

\(\displaystyle c=\sqrt{32}\)

\(\displaystyle c=\sqrt{16*2}=\sqrt{16}\, \times \sqrt{2} = 4\sqrt{2}\)

2. If you know that ALL squares can be made into two special right triangles such that their angles are 45-45-90, then there's a formula you could use:

Let's say that your side length of the square is "a". Then the diagonal of the square (or the hypotenuse of the right triangle) will be \(\displaystyle a\sqrt{2}\).

So using this with a=4:

\(\displaystyle 4\sqrt{2}\)

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:

\(\displaystyle side^2 + side^2 = hypotenuse^2\)

\(\displaystyle 12^2+12^2=hypotenuse^2\)

\(\displaystyle 144 + 144=hypotenuse^2\)

\(\displaystyle \sqrt{144+144}=\sqrt{hypotenuse^2}\)

\(\displaystyle hypotenuse=\sqrt{2\times 144}=\sqrt{2\times 12\times 12}=12\sqrt{2}\)

From the perimeter, we can find the length of each side of the square. The side lengths of a square are equal by definition therefore, the perimeter can be rewritten as,

\(\displaystyle 4s=28\)

\(\displaystyle s=7\)

Then we use the Pythagorean Theorme to find the diagonal, which is the hypotenuse of a right triangle with each leg being a side of the square.

\(\displaystyle a^2+b^2=c^2\)

\(\displaystyle 7^2+7^2=c^2\)

\(\displaystyle c=\sqrt{49+49}=\sqrt{98}=\sqrt{2\cdot49}=7\sqrt2\)

The diagonal of a square is also a hypotenuse of a right triangle with the side lengths as legs of the triangle.

Use the Pythagorean Theorem to find the length of the diagonal.

\(\displaystyle \text{Diagonal}=\sqrt{\text{side length}^2+\text{side length}^2}\)

\(\displaystyle \text{Diagonal}=\sqrt{2(\text{side length})^2}\)

\(\displaystyle \text{Diagonal}=(\text{side length})\sqrt2\)

For the square given in the question,

\(\displaystyle \text{Diagonal}=4\sqrt2\)

The diagonal of a square is also a hypotenuse of a right triangle with the side lengths as legs of the triangle.

Use the Pythagorean Theorem to find the length of the diagonal.

\(\displaystyle \text{Diagonal}=\sqrt{\text{side length}^2+\text{side length}^2}\)

\(\displaystyle \text{Diagonal}=\sqrt{2(\text{side length})^2}\)

\(\displaystyle \text{Diagonal}=(\text{side length})\sqrt2\)

For the square given in the question,

\(\displaystyle \text{Diagonal}=10\sqrt2\)

The diagonal of a square is also a hypotenuse of a right triangle with the side lengths as legs of the triangle.

Use the Pythagorean Theorem to find the length of the diagonal.

\(\displaystyle \text{Diagonal}=\sqrt{\text{side length}^2+\text{side length}^2}\)

\(\displaystyle \text{Diagonal}=\sqrt{2(\text{side length})^2}\)

\(\displaystyle \text{Diagonal}=(\text{side length})\sqrt2\)

For the square given in the question,

\(\displaystyle \text{Diagonal}=15\sqrt2\)

The diagonal of a square is also a hypotenuse of a right triangle with the side lengths as legs of the triangle.

Use the Pythagorean Theorem to find the length of the diagonal.

\(\displaystyle \text{Diagonal}=\sqrt{\text{side length}^2+\text{side length}^2}\)

\(\displaystyle \text{Diagonal}=\sqrt{2(\text{side length})^2}\)

\(\displaystyle \text{Diagonal}=(\text{side length})\sqrt2\)

For the square given in the question,

\(\displaystyle \text{Diagonal}=3\sqrt2\)

The diagonal of a square is also a hypotenuse of a right triangle with the side lengths as legs of the triangle.

Use the Pythagorean Theorem to find the length of the diagonal.

\(\displaystyle \text{Diagonal}=\sqrt{\text{side length}^2+\text{side length}^2}\)

\(\displaystyle \text{Diagonal}=\sqrt{2(\text{side length})^2}\)

\(\displaystyle \text{Diagonal}=(\text{side length})\sqrt2\)

For the square given in the question,

\(\displaystyle \text{Diagonal}=22\sqrt2\)

The diagonal of a square is also a hypotenuse of a right triangle with the side lengths as legs of the triangle.

Use the Pythagorean Theorem to find the length of the diagonal.

\(\displaystyle \text{Diagonal}=\sqrt{\text{side length}^2+\text{side length}^2}\)

\(\displaystyle \text{Diagonal}=\sqrt{2(\text{side length})^2}\)

\(\displaystyle \text{Diagonal}=(\text{side length})\sqrt2\)

For the square given in the question,

\(\displaystyle \text{Diagonal}=12\sqrt2\)