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Test PrepACTMathCircles

ACT Math Skill Review: Circles

Ahh, the circle. You might wax poetic about it, but it trips a lot of people up when it appears on the ACT. Let’s take a look at a circle and then review its basic facts:

Circle

Circle Facts

  • The diameter of a circle is a line segment that passes through the center (O) and has its endpoints on the circle (as in BOC above). All diameters of the same circle have equal length.
  • The radius of a circle is a line segment extending from the center of the circle to a point on the circle. All radii of the same circle have equal length (OA, OB and OC are all radii). The radius is equal to half of the diameter.
  • A chord is a line connecting two points on a circle (Line segment AC above is a chord). 
  • An arc is part of a circle. An arc can be measured in degrees or in units of length. AB is an arc. If you form an angle by drawing radii from the ends of the arc to the center of the circle, the number of degrees in the arc equals the number of degrees in the angle formed. 
  • A tangent to a circle is a line that touches the circle at only one point. A tangent is always perpendicular to the radius that contains the point of the line that touches the circle. 
  • The circumference is the distance around a circle. It is equal to 2πr where r is the radius or πD, where D is the diameter. 
  • The area of a circle is equal to πr2.

Examples


Answers and Explanations

  1. The correct answer is A. First thing, draw in the missing line connecting A, B and C. This is the radius of the larger circle and the diameter of the smaller circle. Now look at the other piece of information the question stem gives us: the diameter of the larger circle is 10 (note: this is choice E!). That means the radius of the larger circle is 5. So, the length of ABC is 5 (note: this is choice C!). This is also the diameter of the smaller circle. But, that’s not what we are looking for; we need the radius of the smaller circle, which is half of its diameter or, 2.5. This is choice A. It’s easy to trip up and pick C if you aren’t careful! Consider circling the information you need to find—in this case the radius of the smaller circle—so you don’t forget it!
     
  2. The correct answer is H. If the diameter of the circle is 6, we know the radius is 3. So, both AC and BO equal 3. Write this on your diagram. The question stem also tells you that angle AOB is 60 degrees, so write that piece of information on your diagram as well. Since you are looking for the length of line segment AB, draw the line connecting points A and B. Voila! At this point, you are finished with the circle portion of the question and you can move on to the triangle stage! Indeed, you should notice that since AO and BO are both equal in length, they must be opposite angles of equal measure. Thus, the remaining two angles of the triangle must also equal 60 degrees and the triangle is equilateral. From there, you should remember that all the sides of an equilateral triangle are equal, so line segment AB must also equal 3. Choice I is actually the measure of minor arc AB–so if you were tempted by that choice, review your circle terminology!


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