Geometry
Circles
After so many cruel, rigid line drawings, it’s nice to contemplate the circle, nature’s roundest creation:
This is a circle with center C. All points along the edge are equidistant from C. This means that line segments . The equals sign with a squiggle over it is the symbol for congruence, the geometrical way of saying that the three line segments are all equal. Any line going from the center of the circle to the edge is called a radius. , , and are all radii. If a line goes from the edge, through the center, and continues on to the other edge, it is called a diameter, and its length is equal to two radii. The diameter bisects a circle. is a diameter.
The circumference of a circle is the distance around the outer edge, so circumference is like saying “perimeter of a circle.” The circumference and area formulas for a circle are:
Both of these formulas contain the mathematical constant pi, which has a value of around 3.14 and looks like this: , and the variable r for radius (in the circumference formula you can also use the variable d for diameter, which is equal to 2r). This means that if you’re given the circumference of a circle, you can find its area by first solving for r using the circumference formula and then plugging that value of r into the area formula. You can use the area of a circle to find its circumference by doing the same thing: solving for r and then placing that value into the proper formula. Test-makers think this is very clever, so you should expect to do it on the SAT.
Tangent Lines
Properties of tangent lines are one of the concepts covered on the new SAT. Look at line GJ. It touches the circle at only one point, F. At the point where a tangent touches a circle, the tangent line is perpendicular to the circle.
Whenever a tangent hits a circle, it creates a right angle with a radius of the circle. Earn some brownie points and think of what the SAT could do with this fact. Draw a dotted line from G to C, and draw another from J to C. What you get is two right triangles.
Arcs, Sectors, and Some Other Concepts That Sound Harder Than They Are
It’s slice-and-dice time!
Angle DEF has its vertex at the center of circle E, so it shouldn’t blow your mind that this is called a central angle. Since is a right angle, the arc (portion of the circle) DE sliced out by this angle is equal to 90˚. An arc defined by a central angle is always equal to the measure of that angle.
Contrast this to an inscribed angle, which is an angle formed by two chords (lines from one point on the circle to another that do not pass through the center) with a vertex resting on the edge of the circle. Angle DGF is an inscribed angle formed by chords GD and GF. An inscribed angle is half the size of the arc it creates. In our diagram above, we already know that arc DF is 90˚. Therefore, angle DGF must be half of 90˚, so . This little fact makes inscribed angles quite sneaky, so if you do encounter one, it will probably be on a medium or hard item.
Arc lengths and sectors are pretty straightforward concepts:
Arc length = portion of the circumference.
Sector = portion of the area.
You don’t have to be Stephen Hawking to realize that before you solve for arc length, you have to solve for the circumference. Circle E has a radius of 5, so the circumference is:
Now, arc DF takes up 90˚, and 90˚ is one-fourth of the total degrees in a 360˚ circle, so (drumroll) . . . arc length DF will be one-fourth of the total circumference.
Sector is the area equivalent of arc length. Look at the shaded portion of the circle, which takes up one-fourth of the total circle since the central angle is 90˚. To find the area of the shaded sector, find the area and multiply it by whatever fraction of the total circle the sector comprises.
Arc lengths and sectors are not basic geometry concepts, and their weirdness is what makes them tough for many students. But their bark is worse than their bite: once you learn what they mean, finding the correct value simply requires multiplying the total area or circumference by a fraction of the total circle.
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