Problem 2
Select all quantities that are proportional to the diagonal length of a square.
A. Area of a square
B. Perimeter of a square
C. Side length of a square

Question

Problem 2 Select all quantities that are proportional to the diagonal length of a square. A. Area of a square B. Perimeter of a square C. Side length of a square

UpStudy AI · Accepted Answer

Let the side length of the square be $$ s $$.

1. The diagonal length $$ d $$ of the square is given by 
   $$
   d = s\sqrt{2}.
   $$
   
2. **Area of a square:**  
   The area is 
   $$
   A = s^2.
   $$
   Express $$ A $$ in terms of $$ d $$:
   $$
   s = \frac{d}{\sqrt{2}} \quad \Longrightarrow \quad A = \left(\frac{d}{\sqrt{2}}\right)^2 = \frac{d^2}{2}.
   $$
   This shows that the area is proportional to $$ d^2 $$, not directly proportional to $$ d $$.

3. **Perimeter of a square:**  
   The perimeter is 
   $$
   P = 4s.
   $$
   Substitute $$ s $$ in terms of $$ d $$:
   $$
   P = 4\left(\frac{d}{\sqrt{2}}\right) = \frac{4d}{\sqrt{2}} = 2\sqrt{2}\,d.
   $$
   Thus, the perimeter is directly proportional to $$ d $$.

4. **Side length of a square:**  
   We already have 
   $$
   s = \frac{d}{\sqrt{2}},
   $$
   which shows that the side length is directly proportional to $$ d $$.

Hence, the quantities proportional to the diagonal length of a square are the perimeter and the side length.

The correct selections are **B. Perimeter of a square** and **C. Side length of a square**.
The perimeter and side length of a square are proportional to its diagonal length.

Problem 2
Select all quantities that are proportional to the diagonal length of a square.
A. Area of a square
B. Perimeter of a square
C. Side length of a square

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