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 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

• 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 either statement (1) alone or statement (2) alone is sufficient to answer the question

Hint:

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.

Answer 1

The Correct Option is D

Step 1: Identify what kind of triangle ABC is.
A, B, C lie on the outer circle, so that circle is the circumcircle of triangle ABC, with circumradius \(R\). Both AB and AC are tangent to the inner circle, so the inner circle behaves like the incircle of the triangle at vertex A, with inradius \(r\). We are told the outer circle's area is four times the inner circle's area, so \(\pi R^2=4\pi r^2\), giving \(R=2r\). By the relation between circumradius, inradius and the distance between their centers, \(OI^2=R(R-2r)\); when \(R=2r\) this distance becomes zero, meaning the circumcenter and incenter coincide. That only happens for an equilateral triangle. So triangle ABC is equilateral, and its area depends only on \(R\) through \[ \text{Area}=\frac{3\sqrt{3}}{4}R^2 \] So only one solid number for \(R\) (or for the outer circle's area, which fixes \(R\)) is needed from the statements.

Step 2: Check statement (1) alone.
The outer circle's area is 12 sq cm, so \(\pi R^2=12\), meaning \(R^2=\frac{12}{\pi}\). Substituting into the area formula gives \[ \text{Area}=\frac{3\sqrt{3}}{4}\times\frac{12}{\pi}=\frac{9\sqrt{3}}{\pi}\text{ sq cm} \] a single fixed value, so statement (1) alone is enough.

Step 3: Check statement (2) alone.
The region between the two circles (the ring) has area 9 sq cm, that is, outer area minus inner area = 9. Since outer area is 4 times inner area, this becomes \(4\times(\text{inner area})-(\text{inner area})=9\), so \(3\times(\text{inner area})=9\), giving inner area = 3 sq cm and outer area = 12 sq cm. This is exactly the same outer area as in statement (1), so the same computation gives \(\text{Area}=\frac{9\sqrt{3}}{\pi}\) sq cm, using only statement (2).

Step 4: Final answer.
Each statement, on its own, lets us find the outer circle's area and hence the exact area of triangle ABC. \[ \boxed{\text{Either statement alone is sufficient}} \]