Difficulty: Easy
Correct Answer: 240 sq cm
Explanation:
Introduction / Context:
This question uses the Pythagoras theorem to find the unknown side of a rectangle when the diagonal and one side are known. Once both sides of the rectangle are known, the area is simply the product of length and breadth. The rectangle can be viewed as a right triangle with the diagonal as hypotenuse and the two sides as perpendicular legs.
Given Data / Assumptions:
Concept / Approach:
Let the unknown side be x cm. Then by the Pythagoras theorem, d^2 = 10^2 + x^2. Substitute d = 26 and solve for x. Once x is found, area = 10 * x. This approach directly uses the right triangle formed by the sides and diagonal of the rectangle.
Step-by-Step Solution:
Let known side = 10 cm and unknown side = x cm.Diagonal d = 26 cm.By Pythagoras: d^2 = 10^2 + x^2.So 26^2 = 100 + x^2.676 = 100 + x^2.x^2 = 676 - 100 = 576.x = sqrt(576) = 24 cm.Area of rectangle = 10 * 24 = 240 sq cm.
Verification / Alternative check:
Check diagonal again: sqrt(10^2 + 24^2) = sqrt(100 + 576) = sqrt(676) = 26 cm, exactly as given. Thus the sides 10 and 24 are correct. The area 240 sq cm is consistent with these dimensions and lies between 10^2 and 26^2, which is reasonable.
Why Other Options Are Wrong:
120 sq cm results from halving the correct area or using only one side squared. 360 sq cm and 480 sq cm come from misusing the diagonal as one of the sides or from arithmetic mistakes. 260 sq cm is a near miss often produced by adding 10 and 24 instead of multiplying them. None of these incorrect values matches the consistent right triangle relationship.
Common Pitfalls:
Missing the fact that the diagonal and sides of a rectangle form a right triangle is a major error. Some students misapply the formula by writing 26 = 10 + x, which is linear, instead of 26^2 = 10^2 + x^2. Others take the square root at the wrong stage. Clear understanding of the Pythagoras theorem prevents these mistakes.
Final Answer:
240 sq cm
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