A Question is given followed by two Statements I and II. Consider the Question and the Statements and mark the correct option. Question: ABC is a triangle right-angled at A and AD is perpendicular to BC. If AD = \(7.2\) cm, then what is \(BD \times CD\) equal to? Statement I: \(AB : AC = 3 : 4\). Statement II: \(BC = 15\) cm. Which one of the following is correct in respect of the above Question and Statements?
The Question can be answered even without using both the Statements.
In a right triangle right-angled at A with altitude AD drawn to the hypotenuse BC, triangles ABD, CAD and CBA are all similar, which gives the standard geometric mean relation \(AD^2 = BD \times CD\). This relation depends only on the length of the altitude AD and holds irrespective of the individual leg lengths, so it does not require the ratio \(AB:AC\) (Statement I) or the length of \(BC\) (Statement II).
Substituting \(AD = 7.2\) cm gives \(BD \times CD = (7.2)^2 = 51.84\) cm², which is already determined from the Question itself. Hence the correct option is (d).
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