Geometry - How to find the length of the diagonal of a square

Find the length of the square's diagonal.

Square_8

$8\sqrt2$

$4\sqrt5$

None of the other answers are correct.

Explanation

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: , 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:

$c=\sqrt{128}=\sqrt{2\cdot 64}=8\sqrt2$

The perimeter of a square is 48. What is the length of its diagonal?

$12\sqrt{2}$

$\sqrt{12}$

$24\sqrt{2}$

$48\sqrt{2}$

$2\sqrt{12}$

Explanation

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:

$\sqrt{144+144}=\sqrt{hypotenuse^2}$

$hypotenuse=\sqrt{2\times 144}=\sqrt{2\times 12\times 12}=12\sqrt{2}$

Find the length of the diagonal of a square that has side lengths of cm.

$4\sqrt{2}$ $\text{cm}$

$16\text{ cm}$

$4\text{ cm}$

$25\text{ cm}$

$10\text{ cm}$

Explanation

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: , where a and b are the side lengths of the square and c is the length of the diagonal.

$c=\sqrt{32}$

$c=\sqrt{16*2}=\sqrt{16}, \times \sqrt{2} = 4\sqrt{2}$

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 $a\sqrt{2}$ .

So using this with a=4:

$4\sqrt{2}$

True or false: The length of a diagonal of a square with sides of length 1 is $\sqrt{3}$ .

False

True

Explanation

A square is shown below with its diagonal.

Square 1

Each of the triangles formed is an isosceles right triangle with congruent legs - by the 45-45-90 Triangle Theorem, they are 45-45-90 triangles. Also by the 45-45-90 Triangle Theorem, the diagonal, the hypotenuse of each triangle, measures $\sqrt{2}$ times the length of a leg. Since each side of the square measures 1, the diagonal has length $\sqrt{2}$ , not $\sqrt{3}$ .

Find the length of the diagonal of a square whose side length is 3.

$3\sqrt2$

Explanation

To find a diagonal of a square recall that the diagonal will create a triangle in the square for which it is the hypotenuse and the side lengths will be the other two lengths of the triangle.

To solve, simply use the Pythagorean Theorem to solve.

Thus,

$c=\sqrt{a^2+b^2}=\sqrt{3^2+3^2}=\sqrt{18}=3\sqrt2$

Find the length of the diagonal of a square whose side length is .

$8\sqrt{2}$

$16\sqrt{2}$

Explanation

To find the diagonal, you case use the pythagorean theorem or realize that this in isosceles triangle, and therefore the hypotenuse is

$s\sqrt{2}=8\sqrt{2}$

Side in the square below has a length of 12. What is the length of the diagonal ?

Square_diagonal

$12\sqrt{2}$

Cannot be determined from information given.

Explanation

Diagonal forms a triangle with adjacent sides . Since this is a square we know this is a right triangle and we can use the Pythagorean Theorem to determine the length of . Sides of length form each of the legs and is the hypotenuse. So the equation looks like this:

$12^{2}+12^{2}=d^{2}$

Solve for

$144+144=d^{2}$

$288=d^{2}$

$d=\sqrt{288}$

We can simplify this to

$\sqrt{144\cdot 2}=\sqrt{144}\cdot \sqrt{2}=12\sqrt{2}$

Find the length of the diagonal of a square with side length 2.

$2\sqrt2$

$3\sqrt2$

Explanation

To solve, simply use the Pythagorean Theorem. Thus,

$c=\sqrt{a^2+b^2}=\sqrt{2^2+2^2}=\sqrt{2(2^2)}=2\sqrt{2}$

Remember, that when simplifying square roots, you can only pull a number out if you have two factors of it. That is why I grouped the end of this answer the way I did so you could see that since I had two squared, I could pull one out, one disappears, and the two on the outside of the parenthesis remains under the radical.

Find the length of the diagonal of a square with side lengths of .