Question:hard

The perimeter of a right angled triangle ABC, right angled at A, is \(3+\sqrt{3}\) cm. What is the area of the triangle?
Statement 1: \(AC \neq AB\)
Statement 2: \(\angle ABC = 30^{\circ}\)

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A right triangle with one acute angle fixed (here 30 degrees) has its sides in the fixed ratio \(1:\sqrt{3}:2\); combine that with the given perimeter to find the actual side lengths.
Updated On: Jul 21, 2026
  • If the data in statement (1) alone is sufficient to answer the question
  • If the data in statement (2) alone is sufficient to answer the question
  • If the data in both the statements together are needed to answer the question
  • If neither statement (1) nor statement (2) suffices to answer the question
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The Correct Option is D

Solution and Explanation

Step 1: Set up the triangle using the right angle at A.
In triangle ABC, angle A is 90 degrees, so BC is the hypotenuse and AB, AC are the two legs. We are told the perimeter \(AB+AC+BC = 3+\sqrt{3}\) cm and asked to find the area, which is \(\frac{1}{2}\times AB \times AC\).

Step 2: Test statement (1) on its own.
\(AC \neq AB\) simply rules out the special case where the triangle is a 45-45-90 triangle. It puts no other restriction on the legs. Many different unequal pairs of legs can add up with the hypotenuse to \(3+\sqrt{3}\), and each pair gives a different product \(AB \times AC\), hence a different area. So the area cannot be pinned to one number from this statement alone.

Step 3: Test statement (2) on its own using trigonometric ratios.
With \(\angle ABC = 30^{\circ}\) and \(\angle A = 90^{\circ}\), \(AC = BC\sin 30^{\circ} = \frac{BC}{2}\) and \(AB = BC\cos 30^{\circ} = \frac{BC\sqrt{3}}{2}\). Substituting into the perimeter equation: \[ \frac{BC}{2}+\frac{BC\sqrt{3}}{2}+BC = 3+\sqrt{3} \] \[ BC\cdot\frac{3+\sqrt{3}}{2}=3+\sqrt{3} \] which gives \(BC = 2\) cm. Then \(AC=1\) cm and \(AB=\sqrt{3}\) cm, and the area comes out as \(\frac{1}{2}\times\sqrt{3}\times1=\frac{\sqrt{3}}{2}\) sq cm, a single fixed number.

Step 4: Compare and conclude.
Statement (2) alone locks the triangle to one shape and size and gives a unique area of \(\frac{\sqrt{3}}{2}\) sq cm, while statement (1) alone still allows many different areas. \[ \boxed{\text{Statement (2) alone is sufficient}} \]
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