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If line segments AB, BC and CD are equal in length in the figure shown and A is the center of the circle, what is the value of x?

A) 15
B) 30
C) 45
D) 60
E) 75

We know that AB, BC and CD are equal in length. Since AB is a radius of the circle and AD and AC are also radii of the circle, we know that AD and AC are also equal in length to AB and the others.

Since all those distances are equal, what we have here includes two equilateral triangles, ABC and ADC, and that helps A LOT, because all three of the interior angles of equilateral triangles measure 60 degrees.

So angle BAC is 60 degrees and angle DAC is 60 degrees.

We can add angles BAC and DAC to get angle BAD, 60 + 60 = 120, and we are almost done.

Since AB = AD, triangle BAD is isosceles, with one angle, BAD, measuring 120 degrees, meaning the other two angles are equal and add up to 60 degrees. So angle x = 60/2 = 30.

Choose B.

Marty Murray wrote:
image hosting

If line segments AB, BC and CD are equal in length in the figure shown and A is the center of the circle, what is the value of x?

A) 15
B) 30
C) 45
D) 60
E) 75

We know that AB, BC and CD are equal in length. Since AB is a radius of the circle and AD and AC are also radii of the circle, we know that AD and AC are also equal in length to AB and the others.

Since all those distances are equal, what we have here includes two equilateral triangles, ABC and ADC, and that helps A LOT, because all three of the interior angles of equilateral triangles measure 60 degrees.

So angle BAC is 60 degrees and angle DAC is 60 degrees.

We can add angles BAC and DAC to get angle BAD, 60 + 60 = 120, and we are almost done.

Since AB = AD, triangle BAD is isosceles, with one angle, BAD, measuring 120 degrees, meaning the other two angles are equal and add up to 60 degrees. So angle x = 60/2 = 30.

Choose B.

Thanks! Thats a nice explanation.