A Question is given followed by two Statements I and II. Consider the Question and the Statements and mark the correct option. Question: ABCD is a quadrilateral with \(AB = 6\) cm, \(BC = 8\) cm, \(CD = 4\sqrt{5}\) cm, \(DA = 2\sqrt{5}\) cm and inscribed in a circle. What is the diameter of the circle? Statement I: The area of the quadrilateral is \(44\) cm². Statement II: \(\angle ABC = 90^\circ\). Which one of the following is correct in respect of the above Question and Statements?
The Question can be answered by using either Statement alone.
Since ABCD is cyclic, opposite angles are supplementary: \(\angle ADC = 180^\circ - \angle ABC\), so \(\sin \angle ADC = \sin \angle ABC\). Splitting the quadrilateral along diagonal AC, its area equals \(\tfrac{1}{2}\,AB \cdot BC \cdot \sin\angle ABC + \tfrac{1}{2}\,AD \cdot DC \cdot \sin\angle ADC = \tfrac{1}{2}\sin\angle ABC\,(AB\cdot BC + AD\cdot DC)\).
Here \(AB\cdot BC = 6\times 8 = 48\) and \(AD\cdot DC = 2\sqrt5\times4\sqrt5 = 40\), so Area \(= \tfrac12\sin\angle ABC\,(48+40)=44\sin\angle ABC\).
Using Statement I alone: Area \(=44\) cm² \(\Rightarrow \sin\angle ABC = 1 \Rightarrow \angle ABC = 90^\circ\). Since ABCD is cyclic, the chord AC subtending a right angle at B must be a diameter, so diameter \(= AC = \sqrt{AB^2+BC^2} = \sqrt{36+64} = \sqrt{100} = 10\) cm. Statement I alone is sufficient.
Using Statement II alone: \(\angle ABC = 90^\circ\) directly means AC subtends a right angle at B, so AC is a diameter: diameter \(= \sqrt{6^2+8^2} = 10\) cm. Statement II alone is also sufficient.
Since either Statement alone suffices, the correct option is (b).
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A Question is given followed by two Statements I and II. Consider the Question and the Statements and mark the correct option.
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