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 to answer the question asked.
• B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
• C. BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.
• D. EACH statement ALONE is sufficient to answer the question asked.
• E. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

Hint:

For geometry problems asking for perimeter or area, look for algebraic identities that connect the given information. The identity \((l+w)^2 = l^2 + w^2 + 2lw\) is very useful for rectangle problems when the diagonal (\(l^2+w^2\)) and area (\(lw\)) are involved.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
We need to find the perimeter of a rectangle. Let the length of the rectangle be \(l\) and the width be \(w\).
The perimeter \(p\) is given by the formula \(p = 2(l+w)\). To find a unique value for \(p\), we need to find a unique value for the sum \((l+w)\).
Step 2: Key Formula or Approach:
For a rectangle:

Area: \(A = l \times w\)
Diagonal (by Pythagorean theorem): \(d^2 = l^2 + w^2\)
Algebraic identity: \((l+w)^2 = l^2 + w^2 + 2lw\)
Step 3: Detailed Explanation:
Analyze Statement (1): Each diagonal of rectangle Q has length 10.
This gives us the equation \(l^2 + w^2 = 10^2 = 100\). This single equation has two variables, \(l\) and \(w\), so it can have multiple solutions.

If \(l=8\) and \(w=6\), then \(8^2+6^2 = 64+36 = 100\). The perimeter would be \(p = 2(8+6) = 28\).
If \(l=\sqrt{50}\) and \(w=\sqrt{50}\) (a square), then \((\sqrt{50})^2+(\sqrt{50})^2 = 50+50 = 100\). The perimeter would be \(p = 2(\sqrt{50}+\sqrt{50}) = 4\sqrt{50} = 20\sqrt{2} \approx 28.28\).
Since the perimeter can have different values, statement (1) is not sufficient.
Analyze Statement (2): The area of rectangle Q is 48.
This gives us the equation \(l \times w = 48\). This single equation also has multiple possible solutions for \(l\) and \(w\).

If \(l=8\) and \(w=6\), the area is \(8 \times 6 = 48\). The perimeter would be \(p = 2(8+6) = 28\).
If \(l=12\) and \(w=4\), the area is \(12 \times 4 = 48\). The perimeter would be \(p = 2(12+4) = 32\).
Since the perimeter can have different values, statement (2) is not sufficient.
Analyze Both Statements Together:
We now have a system of two equations with two variables: 1) \(l^2 + w^2 = 100\) 2) \(lw = 48\) We want to find \(p = 2(l+w)\). We can use the algebraic identity \((l+w)^2 = l^2 + w^2 + 2lw\). Substitute the values from our two equations into this identity: \[ (l+w)^2 = (100) + 2(48) \] \[ (l+w)^2 = 100 + 96 = 196 \] Taking the square root of both sides: \[ l+w = \sqrt{196} = 14 \] (We take the positive root since length and width must be positive). Now we can find the perimeter: \[ p = 2(l+w) = 2(14) = 28 \] Since we have found a unique value for the perimeter, the statements together are sufficient.
Step 4: Final Answer:
Neither statement alone is sufficient, but both statements together are sufficient.