Question

Rectangle ABCD has length \( x \) cm, breadth \( y \) cm and diagonal \( z \) cm. What is the perimeter of ABCD? Statement (I): \( x - y = 7 \)
Statement (II): \( z = 13 \)

• A. Statement (I) alone is sufficient.
• B. Statement (II) alone is sufficient.
• C. Both statements (I) and (II) are sufficient.
• D. Neither statement is sufficient.

Hint:

For problems involving rectangle perimeters and diagonals, remember the connection between \( (x+y)^2 \) and \( (x-y)^2 \). If you know the diagonal and the difference of the sides, you can always find the sum of the sides.

Answer 1

The Correct Option is C

Concept: The perimeter \( P \) of a rectangle is defined as: \[ P = 2(x + y) \] In a rectangle, the relationship between sides and diagonal is given by the Pythagorean theorem: \[ x^2 + y^2 = z^2 \]

Step 1: Analyze individual statements. Statement (I) gives \( x - y = 7 \). This provides a relation between \( x \) and \( y \), but we cannot determine \( x + y \) without knowing the value of the diagonal \( z \). Statement (II) gives \( z = 13 \). This gives \( x^2 + y^2 = 169 \), which is insufficient to determine the perimeter on its own.

Step 2: Combine the statements. We have the system of equations: \[ x - y = 7 \] \[ x^2 + y^2 = 169 \]

Step 3: Solve for the perimeter \( 2(x + y) \). Recall the algebraic identity: \[ (x - y)^2 = x^2 + y^2 - 2xy \] Substitute the known values: \[ 7^2 = 169 - 2xy \] \[ 49 = 169 - 2xy \implies 2xy = 120 \] Now use the identity for \( (x + y)^2 \): \[ (x + y)^2 = x^2 + y^2 + 2xy \] Substitute the known values: \[ (x + y)^2 = 169 + 120 = 289 \] Taking the square root: \[ x + y = \sqrt{289} = 17 \] The perimeter is: \[ P = 2(x + y) = 2(17) = 34 \text{ cm} \] 34 { cm