To determine the length of the diagonal of a square given the side length of 16 units, you can use the Pythagorean theorem, which states that in a right-angled triangle, the square of the length of the hypotenuse (diagonal in this case) is equal to the sum of the squares of the lengths of the other two sides (both sides of the square).
Here's the step-by-step solution:
1. Identify the side length of the square: The given side length is 16 units.
2. Understand the relationship in a square: In a square, the diagonal splits the square into two right-angled triangles.
3. Apply the Pythagorean theorem:
- For a square with side length [tex]\( a \)[/tex]:
[tex]\[
\text{diagonal}^2 = \text{side}^2 + \text{side}^2
\][/tex]
- Substitute the given side length (16 units):
[tex]\[
\text{diagonal}^2 = 16^2 + 16^2
\][/tex]
4. Calculate the squares of the sides:
[tex]\[
16^2 = 256
\][/tex]
[tex]\[
\text{So, } \text{diagonal}^2 = 256 + 256 = 512
\][/tex]
5. Find the square root of 512 to get the diagonal length:
[tex]\[
\text{diagonal} = \sqrt{512}
\][/tex]
6. Simplify the square root of 512:
[tex]\[
\sqrt{512} = \sqrt{256 \times 2} = \sqrt{256} \times \sqrt{2} = 16 \times \sqrt{2}
\][/tex]
7. Calculate the numerical value of the diagonal:
With [tex]\( \sqrt{2} \)[/tex] approximately equal to 1.414,
[tex]\[
16 \times 1.414 = 22.627416997969522
\][/tex]
Thus, the length of the diagonal of the square with side length 16 units is approximately 22.627416997969522 units.
