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Circles

A circle is a set of points in a plane equidistant from a fixed point (the center of the circle). The perimeter of a circle is called the circumference.



A line segment from a circle to its center is a radius.

A line segment with both end points on a circle is a chord.

A chord passing through the center of a circle is a diameter.

A diameter can be viewed as two radii, and hence a diameter’’s length is twice that of a radius.

A line passing through two points on a circle is a secant.

A piece of the circumference is an arc.

The area bounded by the circumference and an angle with vertex at the center of the circle is a sector.


A tangent line to a circle intersects the circle at only one point. The radius of the circle is perpendicular to the tangent line at the point of tangency:



AB = AC

Two tangents to a circle from a common exterior point of the circle are congruent:





An angle inscribed in a semicircle is a right angle:




A central angle has by definition the same measure as its intercepted arc:




An inscribed angle has one-half the measure of its intercepted arc:




The area of a circle is πr2 , and its circumference (perimeter) is 2πr, where r is the radius:


A = πr2
C = 2πr

 

In the GRE, π = 3 is a sufficient approximation for π. You don’’t need π = 3.14.

Example:
In the figure below, the circle has center O and its radius is 2. What is the length of arc ACB ?



(A) π/3
(B) 2π/3
(C) π
(D) 4π/3
(E) 7π/3

 

The circumference of the circle is 2πr = 2π(2) = 4π. A central angle has by definition the same degree measure as its intercepted arc. Hence, arc ACB is also 60°. Now, the circumference of the circle has 360°. So arc ACB is 16 (= 60/360) of the circle’’s circumference. Hence, arc ACB = 16 (4π) = 23 π. The answer is (B).




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