The diagonal of a rectangle is 26 cm and one of its sides is 10 cm. What is the area of the rectangle in square centimetres?

Introduction / Context:
This rectangle problem provides the length of the diagonal and one side and asks for the area. The solution involves using the Pythagoras theorem to find the missing side, since the diagonal forms a right triangle with the sides of the rectangle, and then multiplying the two sides to obtain the area.

Given Data / Assumptions:

• The figure is a rectangle.
• Diagonal length d = 26 cm.
• One side is 10 cm. Let this be the breadth b.
• Let the other side (length) be l cm.
• We use the relation d^2 = l^2 + b^2.

Concept / Approach:
In a rectangle, the diagonal acts as the hypotenuse of a right triangle whose legs are the length and breadth. Using the Pythagoras theorem, we can solve for the unknown side given the diagonal and one side, and then compute the area A = l * b. This is a standard application of right triangle geometry inside a rectangle.

Step-by-Step Solution:
Step 1: Use Pythagoras theorem for the rectangle: d^2 = l^2 + b^2.Step 2: Substitute d = 26 and b = 10: 26^2 = l^2 + 10^2.Step 3: Compute the squares: 676 = l^2 + 100.Step 4: Rearrange to get l^2: l^2 = 676 - 100 = 576.Step 5: Take the square root: l = sqrt(576) = 24 cm.Step 6: Now compute the area: A = l * b = 24 * 10 = 240 square centimetres.

Verification / Alternative check:
Verify the diagonal using the found side lengths: d^2 should equal l^2 + b^2.Compute l^2 + b^2 = 24^2 + 10^2 = 576 + 100 = 676.The square root of 676 is 26, which matches the given diagonal.Thus the side lengths and area are consistent with the given information.

Why Other Options Are Wrong:
An area of 260 square centimetres would require different side lengths that do not satisfy the Pythagoras relation with diagonal 26 cm and side 10 cm.Values like 120 square centimetres or 130 square centimetres are too small, while 180 square centimetres also does not match the product of any side pair consistent with the given diagonal.Only 240 square centimetres corresponds to the correct dimensions 24 cm by 10 cm.

Common Pitfalls:
A common error is to assume that the given side of 10 cm is the hypotenuse, which is incorrect, as the diagonal is always the hypotenuse in a rectangle.Another mistake is to confuse area with perimeter and attempt to use formulas that add sides rather than multiply them.Algebraic mistakes when rearranging 676 = l^2 + 100 can also lead to incorrect values for l.

Final Answer:
The area of the rectangle is 240 sq cm.