Bunuel wrote:

In the figure above, points B and C lie on a circle that is centered on point A. Which of the following pieces of information would alone be sufficient to determine the area of the circle?

I. The perimeter of triangle ABC.

II. The length of line segment BC.

III. The area of triangle ABC.

A. I only

B. II only

C. I and III only

D. I, II, and III

E. None of the above.

Attachment:

CircleTriangleABC.png

As above posters have noted, this is an isosceles right triangle.

Two sides are (equal) radii. Their included angle is 90°.

So angle measures are 45-45-90 and corresponding side lengths are

\(x : x : x\sqrt{2}\).

Which of the options is sufficient to determine the area of the circle?

All we need to find area is radius, which equals is one

leg of the triangle.

I. The perimeter of triangle ABC. SUFFICIENT to determine area.

The perimeter would be

\(2x + x\sqrt{2}\) = some number.

That number is a function of the same \(x\) (albeit with one of them multiplied by

\(\sqrt{2}\)).

One variable, one equation: we can find radius \(x\).

II. The length of line segment BC. SUFFICIENT

BC =

\(x\sqrt{2}\). Divide that length by

\(\sqrt{2}\) to get length of \(x\) radius.

As

septwibowo notes, the only option that contains both Options I and II is D.

Agree (kudos), choose it and move on unless there were lots of time, in which case my perfectionism would hijack better instincts.

III. The area of triangle ABC. SUFFICIENT

Area of a 45-45-90 triangle is

\(\frac{s^2}{2}\), because the legs are the triangle's base and height. (In other words, that formula is equivalent to b * h * 1/2). Solving for \(s\) = solving for radius.

Kudos to

Bunuel for posting about 10,000 questions in a week and for smuggling part of DS into PS.

Answer D

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In the depths of winter, I finally learned

that within me there lay an invincible summer.

-- Albert Camus, "Return to Tipasa"