A Question is given followed by two Statements I and II. Consider the Question and the Statements and mark the correct option. Question: The diagonal of a rectangle ABCD (AB > BC) is \(5\sqrt{2}\) cm. What is its perimeter? Statement I: Length AB is an integer. Statement II: Length BC is an integer. Which one of the following is correct in respect of the above Question and Statements?
The Question can be answered by using both the Statements together, but cannot be answered using either Statement alone.
Since ABCD is a rectangle, \(AB^2 + BC^2 = (5\sqrt{2})^2 = 50\). Using Statement I alone (AB an integer), AB can be 1, 2, ..., 7 with BC taking different non-integer values in general, so the perimeter is not fixed. Similarly Statement II alone is insufficient. Using both together, AB and BC must both be positive integers satisfying \(AB^2+BC^2=50\) with \(AB > BC\): the only integer pairs are (5,5) and (7,1); since AB must be strictly greater than BC, (5,5) is rejected, leaving AB = 7, BC = 1 uniquely. Hence the perimeter \(= 2(7+1) = 16\) cm can be found only by using both Statements together, so option (c) is correct.
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Important Questions from 2-D Mensuration
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- In a circle of radius 10.5 cm, if the angle of a sector is $\frac{2\pi}{3}$, then the perimeter of the sector is (in cm):
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Take $\pi = \frac{22}{7}$
