If the length of the diagonal of a square is 20 cm, then its perimeter must be

### Concept & Diagonal to Perimeter Conversion To find the perimeter from the diagonal, establish the side length first using the diagonal-side relationship, then multiply by 4. $$ d = a\sqrt{2} $$ $$ \text{Perimeter} = 4a $$ ### Step-by-Step Solution * Given the diagonal $d = 20$ cm. * Set up the equation for the side length $a$: $a\sqrt{2} = 20$. * Solve for $a$: $a = \frac{20}{\sqrt{2}}$. * Rationalize the denominator: $a = \frac{20\sqrt{2}}{2} = 10\sqrt{2}$ cm. * Calculate the perimeter: $4a = 4 \times 10\sqrt{2} = 40\sqrt{2}$ cm. ### Exam Strategy & Shortcut You can combine the formulas to create a direct relation: $\text{Perimeter} = 4 \times (\frac{d}{\sqrt{2}}) = 2d\sqrt{2}$. Plugging in $d = 20$ immediately gives $2(20)\sqrt{2} = 40\sqrt{2}$ cm. ### Common Pitfall Students might calculate the side length $10\sqrt{2}$ and mistakenly pick option (a) without proceeding to calculate the final perimeter. ### Final Answer Therefore, the correct answer is **$40\sqrt{2}$ cm**.