Question

The diagonal of a square is \(10\sqrt{2}\) units. What is the side length of the square?

• A. 10 units
• B. 5 units
• C. \(5\sqrt{2}\) units
• D. 15 units

Hint:

For any square, the relationship between the side (s) and diagonal (d) is always \(d = s\sqrt{2}\). If a question gives you the diagonal with a \(\sqrt{2}\) component, the side length is often the number in front of the \(\sqrt{2}\). This is a quick check.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept
The diagonal of a square divides it into two congruent right-angled isosceles triangles. The sides of the square are the legs of the triangles, and the diagonal is the hypotenuse.
Step 2: Key Formula or Approach
Let 's' be the side length of the square and 'd' be the length of the diagonal. Using the Pythagorean theorem (\(a^2+b^2=c^2\)), we have \(s^2 + s^2 = d^2\), which simplifies to \(2s^2 = d^2\). Taking the square root gives the direct formula:
\[ d = s\sqrt{2} \] To find the side length, we can rearrange this to:
\[ s = \frac{d}{\sqrt{2}} \] Step 3: Detailed Explanation
We are given the diagonal length:
d = \(10\sqrt{2}\) units
Using the formula \(d = s\sqrt{2}\), we can substitute the given value for d:
\[ 10\sqrt{2} = s\sqrt{2} \] To solve for 's', divide both sides of the equation by \(\sqrt{2}\):
\[ s = \frac{10\sqrt{2}}{\sqrt{2}} \] \[ s = 10 \text{ units} \] Step 4: Final Answer
The side length of the square is 10 units.