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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 though 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:

 

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

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:
 
On the test, π ≈ 3 is a sufficient approximation for π. You don’t need π ≈ 3.14.
 
Example

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

  1. π/3
  2. 2π/3
  3. π
  4. 4π/3
  5. 7π/3
Solution

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  (= 60/360) of the circle’s circumference.

 

Hence, arc ACB = .

 

The answer is (B).
 





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