Question

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: \(\angle ABC = 30^{\circ}\)

• A. If the data in statement (1) alone is sufficient to answer the question
• B. If the data in statement (2) alone is sufficient to answer the question
• C. If the data in both the statements together are needed to answer the question
• D. If neither statement (1) nor statement (2) suffices to answer the question

Hint:

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.

Answer 1

The Correct Option is D

Step 1: Check statement (1) alone.
Statement (1) only tells us that \(AC \neq AB\), so the triangle is not the isosceles right triangle. The perimeter is fixed at \(3+\sqrt{3}\) cm, but a right triangle with a fixed perimeter still has infinitely many possible pairs of legs as long as the legs are unequal. Each such pair gives a different area, so this statement alone cannot pin down one single area.

Step 2: Check statement (2) alone.
Statement (2) tells us \(\angle ABC = 30^{\circ}\). Since the triangle is right angled at A, angle A is \(90^{\circ}\), so angle C must be \(60^{\circ}\). This makes ABC a fixed 30-60-90 triangle, and in such a triangle the three sides always sit in the ratio \(1 : \sqrt{3} : 2\) (leg opposite 30 degrees, leg opposite 60 degrees, hypotenuse). Taking the sides as \(AC = t\), \(AB = t\sqrt{3}\) and \(BC = 2t\), the perimeter is \(t(1+\sqrt{3}+2) = t(3+\sqrt{3})\). Setting this equal to \(3+\sqrt{3}\) gives \(t = 1\).
So \(AC = 1\) cm, \(AB = \sqrt{3}\) cm and \(BC = 2\) cm, and the area works out to \[ \text{Area} = \frac{1}{2} \times AB \times AC = \frac{1}{2}\times \sqrt{3}\times 1 = \frac{\sqrt{3}}{2}\text{ sq cm} \]
This is one single, fixed value, so statement (2) alone is enough to find the area.

Step 3: Final answer.
Working through the two statements independently, statement (2) by itself fixes the triangle completely and gives area \(\frac{\sqrt{3}}{2}\) sq cm, while statement (1) alone leaves the triangle undetermined. \[ \boxed{\text{Statement (2) alone is sufficient}} \]