Question

If \( p \) is the perimeter of rectangle \( Q \), what is the value of \( p \)?
(1) Each diagonal of rectangle \( Q \) has length 10.
(2) The area of rectangle \( Q \) is 48.

• A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
• B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
• C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
• D. EACH statement ALONE is sufficient.
• E. Statements (1) and (2) TOGETHER are not sufficient

Hint:

When solving for the perimeter of a rectangle, use both the area and the Pythagorean theorem to form a system of equations.

Answer 1

The Correct Option is C

Step 1: Analyze statement (1).
Statement (1) tells us that each diagonal of rectangle \( Q \) has length 10. The formula for the diagonal \( d \) of a rectangle with length \( l \) and width \( w \) is: \[ d = \sqrt{l^2 + w^2} \] Since the diagonal is 10, we have: \[ 10 = \sqrt{l^2 + w^2} \quad \implies \quad 100 = l^2 + w^2 \] This equation alone does not give us enough information to find the perimeter \( p = 2l + 2w \), so statement (1) alone is not sufficient.
Step 2: Analyze statement (2).
Statement (2) tells us that the area of rectangle \( Q \) is 48. The area \( A \) of a rectangle is given by: \[ A = l \times w \] So: \[ l \times w = 48 \] This equation alone does not give us enough information to find the perimeter \( p = 2l + 2w \), so statement (2) alone is not sufficient.
Step 3: Combine both statements.
We now have two equations:
1. \( l^2 + w^2 = 100 \) (from statement (1))
2. \( l \times w = 48 \) (from statement (2))
We can solve this system of equations. Let’s use the identity: \[ (l + w)^2 = l^2 + 2lw + w^2 \] Substituting the known values: \[ (l + w)^2 = 100 + 2 \times 48 = 100 + 96 = 196 \quad \implies \quad l + w = \sqrt{196} = 14 \] Thus, the perimeter is: \[ p = 2l + 2w = 2 \times 14 = 28 \] \[ \boxed{28} \]