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In the figure above, points B and C lie on a circle that is centered

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In the figure above, points B and C lie on a circle that is centered  [#permalink]

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New post 27 Nov 2017, 23:20
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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
CircleTriangleABC.png [ 9.84 KiB | Viewed 732 times ]

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Re: In the figure above, points B and C lie on a circle that is centered  [#permalink]

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New post 28 Nov 2017, 00:58
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Bunuel wrote:
Image
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


Let me try :

First, statement III must be correct.
- Area = base * height * \frac{1}{2}
- From triangle ABC, base and height are same : radius.

Second, statement II also must be correct.
- Triangle ABC is a special triangle : 45 45 90, because it is an isosceles triangle.
- Just know length BC is enough to find another side, therefore we can find the radius.

Because there is no option for statement I & II, we can choose D.

Wdyt?
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Re: In the figure above, points B and C lie on a circle that is centered  [#permalink]

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New post 28 Nov 2017, 02:37
1
First off, this is a right angled isosceles triangle (90,45,45) where AC and AB are equal Let's have AC = b; AB = c; BC = a; and b=c;
Let's plug these into the statements
1. a+b+c = 10 (Constant)
==> sqrt(2b^2)+2b = 10; Gives us the radii as well.
2. a = 10 (Constant). This means 10^2 = 2(b^2) ; which will give us the value of one of the radii (which gives us the area)
3. Area = (b^2)/2; Gives us the information we need.

Hence D
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Re: In the figure above, points B and C lie on a circle that is centered  [#permalink]

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New post 28 Nov 2017, 10:31
Bunuel wrote:
Image
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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Re: In the figure above, points B and C lie on a circle that is centered &nbs [#permalink] 28 Nov 2017, 10:31
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