COMEDK · Maths · 27. Application of Derivatives

The dimensions of the largest rectangle of side \(x\) and \(y\) that can be inscribed in the right angled triangle of sides \(\mathrm{a}\) and \(\mathrm{b}\) is

A \(\dfrac{a}{4}, \dfrac{b}{4}\)
B \(a, b\)
C \(\dfrac{a}{2}, \dfrac{b}{2}\)
D \(\dfrac{3 a}{2}, \dfrac{3 b}{2}\)

Verified Solution

Answer & Solution

Correct Answer

(C) \(\dfrac{a}{2}, \dfrac{b}{2}\)

Step-by-step Solution

Detailed explanation

Let the right-angled triangle have base \(a\) and height \(b\). The rectangle is inscribed such that one vertex lies on the hypotenuse, and two sides lie along the legs of the triangle. Let the rectangle have width \(x\) and height \(y\). Using similar triangles, the small…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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