Question:medium

There are two concentric circles such that the area of the outer circle is four times the area of the inner circle. If A, B and C are three distinct points on the perimeter of the outer circle such that AB and AC are tangents to the inner circle, what is the area of the triangle ABC?
Statement 1: The area of the outer circle is 12 sq cm
Statement 2: The area of the region between the two circles is 9 sq cm

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The condition that the outer area is four times the inner area forces triangle ABC to be equilateral with circumradius R; then area depends only on R.
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 either statement (1) alone or statement (2) alone is sufficient to answer the question
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The Correct Option is D

Solution and Explanation

Step 1: Understanding the Concept.
Since AB and AC both touch the inner circle, that inner circle sits inside angle A, tangent to both its arms, with its center on the bisector of angle A. Since the outer circle's area is exactly four times the inner circle's area, the outer radius is exactly twice the inner radius, \(R=2r\). This particular ratio between circumradius and inradius only happens when the triangle is equilateral, so every angle of triangle ABC is \(60^{\circ}\).

Step 2: Key Formula or Approach.
For an equilateral triangle with circumradius \(R\), the side length is \(a=R\sqrt{3}\), so the area is \[ \text{Area}=\frac{\sqrt{3}}{4}a^2=\frac{\sqrt{3}}{4}\times3R^2=\frac{3\sqrt{3}}{4}R^2 \] So finding the area only needs one number, the outer circle's radius or area.

Step 3: Detailed Explanation.
From statement (1), outer area = 12, so \(R^2=12/\pi\) and \(\text{Area}=\frac{3\sqrt{3}}{4}\times\frac{12}{\pi}=\frac{9\sqrt{3}}{\pi}\) sq cm, a complete answer using statement (1) alone.
From statement (2), the ring area (outer minus inner) is 9. Writing inner area as \(\frac{1}{4}\) of outer area (a fact given in the question), outer \(-\frac{1}{4}\)outer \(=9\), so \(\frac{3}{4}\)outer\(=9\), giving outer area \(=12\), the same number as before, and hence the same triangle area, using statement (2) alone.

Step 4: Final Answer.
Both statements independently pin down the outer circle's area to 12 sq cm, so the triangle's area, \(\frac{9\sqrt{3}}{\pi}\) sq cm, follows from either one alone. \[ \boxed{\text{Either statement alone is sufficient}} \]
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