Question

The perimeter of a right-angled triangle ABC, right angled at A, is (3+√3) cm. What is the area of the triangle?

Statement 1: AC ≠ AB.
Statement 2: ∠ABC = 30°.

• A. If the data in statement (1) alone is sufficient to answer the question, but the data in statement (2) alone is not sufficient.
• B. If the data in statement (2) alone is sufficient to answer the question, but the data in statement (1) alone is not sufficient.
• C. If the data in both the statements together are needed to answer the question.
• D. If either statement (1) alone or statement (2) alone is sufficient to answer the question.
• E. If the data in neither statement (1) nor statement (2) is sufficient to answer the question, and more data is needed.

Hint:

A right triangle's area is fixed once you know the perimeter AND one non-right angle (which fixes the full shape via 30-60-90 side ratios); check what each statement actually fixes.

Answer 1

The Correct Option is B

Since the triangle is right angled at A, BC is the hypotenuse and AB, AC are the two legs. Area = (1/2) x AB x AC. We know the perimeter is (3+√3) cm, but that alone allows many different right triangles of different shapes, so we need a statement that fixes the shape (the angles) as well.

Statement 1 only tells us AC is not equal to AB, i.e. the triangle is not the special isosceles right triangle with angles 45-45-90. This still leaves infinitely many possible right triangles (with different angle combinations) sharing the same perimeter, and each would give a different area. So statement 1 alone is not sufficient.

Statement 2 tells us angle ABC (at vertex B) is 30 degrees. Since angle A is 90 degrees, angle C must be 60 degrees, making this a standard 30-60-90 triangle. In such a triangle the sides opposite the 30, 60 and 90 degree angles are in the ratio 1 : √3 : 2. The side opposite B (30 degrees) is AC, the side opposite C (60 degrees) is AB, and the hypotenuse is BC. Let AC = k, AB = k√3, BC = 2k. Perimeter = k + k√3 + 2k = k(3+√3) = 3+√3, so k = 1. That gives AC = 1 cm, AB = √3 cm, BC = 2 cm (and indeed 1² + (√3)² = 1+3 = 4 = 2², confirming a valid right triangle). Area = (1/2) x √3 x 1 = √3/2 cm², a single definite value. So statement 2 alone is sufficient.

Since statement 2 alone answers the question while statement 1 alone does not, the correct choice is (b).

Note on the provided answer key: the key marks this as (e). However, rechecking the 30-60-90 side ratios against the given perimeter confirms a unique, self-consistent triangle (sides 1, √3, 2, satisfying the Pythagorean relation) and a clean area of √3/2 cm² from statement 2 alone, so our verified answer is (b) with this discrepancy flagged for review.