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Senior RC Moderator V
Joined: 02 Nov 2016
Posts: 4124
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A and B are the end points of the longest line that can be drawn in a  [#permalink]

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Difficulty:   45% (medium)

Question Stats: 66% (02:25) correct 34% (01:58) wrong based on 83 sessions

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A and B are the end points of the longest line that can be drawn in a circle with center X. If C is a point on the circle such that AC = AX = 3, what is the perimeter of triangle ABC?

(A) 9/2
(B) 9
(C) 6 + 3 $$\sqrt{3}$$
(D) 9 + 3 $$\sqrt{3}$$
(E) 9 $$\sqrt{3}$$

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Magoosh GMAT Instructor G
Joined: 28 Dec 2011
Posts: 4472
Re: A and B are the end points of the longest line that can be drawn in a  [#permalink]

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A and B are the end points of the longest line that can be drawn in a circle with center X. If C is a point on the circle such that AC = AX = 3, what is the perimeter of triangle ABC?

(A) 9/2
(B) 9
(C) 6 + 3 $$\sqrt{3}$$
(D) 9 + 3 $$\sqrt{3}$$
(E) 9 $$\sqrt{3}$$

I'm happy to respond. First, "the longest line that can be drawn in a circle" = the diameter. We know AB is a diameter.

X is the center, so AX & BC are radii. AC is a chord that is equal in length to the radius, and XC is another radius, so ACX is an equilateral triangle, with three 60 degree angles. Let's look at a diagram:
Attachment: circle X with triangle ABC.png [ 77.94 KiB | Viewed 1551 times ]

The next piece we need is that any triangle with three vertices on a circle that has a diameter a side is a right triangle. Another way to say that is that any angle that intersects an arc of a semicircle is a right angle. See the "special case" in this blog article:
Inscribed and Circumscribed Circles and Polygons on the GMAT
Thus, ABC is a 30-60-90 triangle. The ratios in this triangle should be familiar. See this blog if they are not:
The GMAT’s Favorite Triangles
We know AC = 3.
We know AB = AX + BX = 6
The third side is the side opposite the 60 degree angle in the 30-60-90 triangle. This is $$\sqrt{3}$$ times the shortest side. Thus:
BC = $$3\sqrt{3}$$

perimeter of ABC = AC + AB + BC = $$9 + 3\sqrt{3}$$

OA = (D)

Does all this make sense?
Mike _________________
Mike McGarry
Magoosh Test Prep

Education is not the filling of a pail, but the lighting of a fire. — William Butler Yeats (1865 – 1939)
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Re: A and B are the end points of the longest line that can be drawn in a  [#permalink]

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_________________ Re: A and B are the end points of the longest line that can be drawn in a   [#permalink] 16 Feb 2019, 02:49
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