A rectangle has a diagonal of 17 cm and a perimeter of 46 cm. Find the area of the rectangle in square centimetres (sq cm).

Introduction:
This question tests geometry involving rectangles, diagonals, and perimeter. A rectangle’s diagonal relates to its sides using the Pythagorean relationship because the diagonal forms a right triangle with the two sides. The perimeter gives a linear relation between the sides. Using both facts together allows us to solve for the two side lengths and then compute the area as length * breadth. The trick is to convert the perimeter into a+b and use (a+b)^2 = a^2 + b^2 + 2ab to find ab directly.

Given Data / Assumptions:

• Diagonal d = 17 cm• Perimeter P = 46 cm• Let sides be a and b (in cm)

Concept / Approach:
Perimeter gives 2(a+b) = 46, so a+b = 23. Diagonal gives a^2 + b^2 = d^2 = 289. Use identity: (a+b)^2 = a^2 + b^2 + 2ab. Since area = ab, we can solve for ab without finding a and b separately.

Step-by-Step Solution:
Step 1: Use perimeter.2(a+b) = 46a + b = 23Step 2: Use diagonal (Pythagorean relation).a^2 + b^2 = 17^2 = 289Step 3: Apply the identity (a+b)^2 = a^2 + b^2 + 2ab.(23)^2 = 289 + 2ab529 = 289 + 2abStep 4: Solve for ab.2ab = 529 - 289 = 240ab = 120Step 5: Area of rectangle = a*b = 120 sq cm

Verification / Alternative check:
We can also find sides explicitly: a+b=23 and ab=120. Solve t^2 - 23t + 120 = 0. Factors are 15 and 8. So sides are 15 cm and 8 cm. Diagonal = sqrt(15^2 + 8^2) = sqrt(225 + 64) = sqrt(289) = 17 cm, which matches. Area = 15*8 = 120 sq cm, confirming the result.

Why Other Options Are Wrong:
110, 130, 140, 150 do not satisfy both the diagonal and perimeter constraints simultaneously.Only 120 corresponds to a valid side pair (15,8) that produces diagonal 17 and perimeter 46.

Common Pitfalls:
• Trying random side pairs without using the identity to get ab directly.• Forgetting that perimeter gives a+b, not ab.• Misusing diagonal as a+b instead of sqrt(a^2+b^2).

Final Answer:
The area of the rectangle is 120 sq cm.