A and B are the end points of the longest line that can be drawn in a

Question

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}

mikemcgarry · Accepted Answer

Dear SajjadAhmad , 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: circle X with triangle ABC.png 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 :-)

HariNadh · Answer

AB is the diameter of the circle. Since X is the centre AX is radius of circle and AC=AX implies AC is also having the length of radius. Radius=3cm

ACX forms an equilateral triangle with sides 3 cm. Joining A and B to C makes angle ACB= 90°(lines joining from the end points of diameter to any other points on the circle makes 90°)

Here AC=3 is base and AB=6 is the hypotenuse. BC= √(6^2-3^2)= 3√3

AB+AC+BC= 3+6+3√3= 9+3√3 (D)

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A and B are the end points of the longest line that can be drawn in a

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A and B are the end points of the longest line that can be drawn in a

Prep Toolkit

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