Question

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 of 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

• 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:

Use R = 2r from the area condition, recognize the tangent configuration forces triangle ABC to be equilateral with circumradius R, then check if each statement alone gives you a numeric value for R (or the circle areas).

Answer 1

The Correct Option is D

Let the outer radius be R and inner radius be r, both circles sharing the same centre O. Area of outer = 4 x area of inner means \(\pi R^2 = 4 \pi r^2\), so \(R = 2r\).

Because A, B, C sit on the outer circle, O is their circumcentre. Because AB and AC are tangent to the inner circle, O is equidistant (distance r) from lines AB and AC, which places O on the bisector of angle BAC. Working out the tangent geometry, the half-angle at A satisfies \(\sin(\theta) = r/R = 1/2\), so \(\theta = 30^\circ\) and angle BAC \(= 60^\circ\). Carrying this through with coordinates (placing A at the top of the circle and reflecting the two tangent lines symmetrically) shows that B and C land exactly \(120^\circ\) apart around the circle from A, making triangle ABC equilateral, and its side BC also turns out to be tangent to the inner circle. In short, whenever R = 2r for two concentric circles, the inner circle behaves like the incircle of the equilateral triangle inscribed in the outer circle as its circumcircle.

For an equilateral triangle with circumradius R, the area works out to \(\frac{3\sqrt{3}}{4}R^2\). So the only extra fact we need is the numerical value of R (or equivalently, the area of either circle).

Statement (1): Area of outer circle = 12, so \(\pi R^2 = 12\), giving \(R^2 = 12/\pi\). Area of triangle \(= \frac{3\sqrt{3}}{4} \times \frac{12}{\pi} = \frac{9\sqrt{3}}{\pi} \approx 4.96\) sq.cm. A definite number, so statement (1) alone is sufficient.

Statement (2): The area between the two circles (outer minus inner) is 9. Since outer = 4 x inner, if inner area = y then outer = 4y, and \(4y - y = 9\), so \(3y = 9\), \(y = 3\), meaning outer area = 12. This is exactly the same outer area as in statement (1), and leads to the same triangle area \(\frac{9\sqrt{3}}{\pi}\) sq.cm. So statement (2) alone is sufficient too.

Since each statement independently pins down the area of the outer circle and hence the area of the triangle, either statement alone is sufficient. The correct choice is option (4).