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If \(AB\) is the diameter of the circle with center \(X\) and \(C\) is a point on the circle such that \(AC = AX = 3,\)

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If \(AB\) is the diameter of the circle with center \(X\) and \(C\) is a point on the circle such that \(AC = AX = 3,\)

by Gmat_mission » Sun Jun 07, 2020 4:05 am

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If \(AB\) is the diameter of the circle with center \(X\) and \(C\) is a point on the circle such that \(AC = AX = 3,\) what is the perimeter of triangle \(ABC?\)

A. \(\dfrac{9\sqrt3}2\)

B. \(9\)

C. \(6+ 3\sqrt3\)

D. \(9+ 3\sqrt3\)

E. \(9\sqrt3\)

[spoiler]OA=D[/spoiler]

Source: Princeton Review
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Re: If \(AB\) is the diameter of the circle with center \(X\) and \(C\) is a point on the circle such that \(AC = AX = 3

by RaghavKhanna » Sun Jun 07, 2020 5:58 pm
Since AB is the diameter, and X is the center
=> AX is the radius.

It is given that AX = 3
=> AB = 6

Also, angles in a semi-circle are 90 degrees.
=> \angle ACB = 90 degrees
So, in \triangle ABC,
AB^2 = AC^2 + BC^2
6^2 = 3^2 + BC^2
BC = 3\sqrt{3}

=> Perimeter of \triangle ABC = 3 + 3 + 3\sqrt{3} = 6 + 3\sqrt{3}
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