The length of the diagonal and the breadth of a rectangle are 25 cm and 7 cm respectively. What is the perimeter of the rectangle in centimetres?

Introduction / Context:
This question concerns a rectangle for which one side length and the diagonal are known. The task is to find the perimeter. The problem tests understanding of the Pythagoras theorem in rectangles and the ability to use it to determine unknown sides, followed by calculating the perimeter from all side lengths.

Given Data / Assumptions:

• The figure is a rectangle.
• Diagonal length d = 25 cm.
• Breadth (width) b = 7 cm.
• Let the length be l cm.
• In a rectangle, the diagonal, length, and breadth form a right triangle.

Concept / Approach:
In any rectangle, the diagonal acts as the hypotenuse of a right triangle whose legs are the length and breadth. The Pythagoras theorem connects these three quantities by d^2 = l^2 + b^2. Once the unknown side l is found using this relation, the perimeter can be calculated using P = 2 * (l + b). This approach uses basic geometry and algebra.

Step-by-Step Solution:
Step 1: Use Pythagoras theorem: d^2 = l^2 + b^2.Step 2: Substitute the known values: 25^2 = l^2 + 7^2.Step 3: Compute the squares: 625 = l^2 + 49.Step 4: Rearrange: l^2 = 625 - 49 = 576.Step 5: Take the square root: l = sqrt(576) = 24 cm.Step 6: Now compute the perimeter: P = 2 * (l + b) = 2 * (24 + 7) = 2 * 31 = 62 cm.

Verification / Alternative check:
Check the diagonal again with l = 24 cm and b = 7 cm.Compute l^2 + b^2 = 24^2 + 7^2 = 576 + 49 = 625.The square root of 625 is 25, which matches the given diagonal.Thus, the side lengths are consistent with the given diagonal, confirming that the perimeter is correctly computed.

Why Other Options Are Wrong:
A perimeter of 41 cm would correspond to a much smaller rectangle and does not match the given diagonal of 25 cm.Perimeters like 82 cm or 124 cm are too large and would require a longer side than 24 cm, which would not satisfy the Pythagoras relation with breadth 7 cm and diagonal 25 cm.A value of 52 cm similarly does not correspond to the side lengths derived from the given diagonal.

Common Pitfalls:
Learners sometimes confuse the perimeter formula with the area formula and try to use l * b instead of 2 * (l + b).Another frequent mistake is misapplying Pythagoras theorem, for example by writing d = l^2 + b^2 instead of d^2 = l^2 + b^2.Rounding errors do not arise here because the numbers form a classic Pythagorean triple 7, 24, 25.

Final Answer:
The perimeter of the rectangle is 62 cm.