Circles
Circles
A circle is the collection of points equidistant from a given point, called the center. A circle is named after its center point. The distance from the center to any point on the circle is called the radius, (r), the most important measurement in a circle. If you know a circle’s radius, you can figure out all its other characteristics. The diameter (d) of a circle is twice as long as the radius (d = 2r) and stretches between endpoints on the circle, passing through the center. A chord also extends from endpoint to endpoint on the circle, but it does not necessarily pass through the center. In the figure below, point C is the center of the circle, r is the radius, and AB is a chord.
Tangent Lines
Tangents are lines that intersect a circle at only one point. Tangents are a new addition to the SAT. You can bet that the new SAT will make sure to cram at least one tangent question into every test.
Just like everything else in geometry, tangent lines are defined by certain fixed rules. Know these rules and you’ll be able to handle anything the SAT throws at you. Here’s the first: A radius whose endpoint is the intersection point of the tangent line and the circle is always perpendicular to the tangent line. See?
And the second rule: Every point in space outside the circle can extend exactly two tangent lines to the circle. The distance from the origin of the two tangents to the points of tangency are always equal. In the figure below, XY = XZ.
Tangents and Triangles
Tangent lines are most likely to appear in conjunction with triangles.
What is the area of triangle QRS if RS is tangent to circle Q?
You can answer this question only if you know the rules of circles and tangent lines. The question doesn’t tell you that QR is the radius of the circle; you just have to know it: Because the circle is named circle Q, point Q must be the center of the circle, and any line drawn from the center to the edge of the circle is the radius. The question also doesn’t tell you that QR is perpendicular to RS. You have to know that they’re perpendicular because QR is a radius and RS is a tangent that meet at the same point.
If you know how to deduce those key facts about this circle, then the actual math in the question is simple. Since QR and RS are perpendicular, and angle RQS is 60°, triangle QRS is a 30-60-90 triangle. The image tells you that side QR, the side opposite the 30° angle equals 4. Side QR is the height of the triangle. To calculate the area, you just have to figure out which of the other two sides is the base. Since the height and base of the triangle must be perpendicular to each other, side RS must be the base. To find RS, use the 1::2 ratio. RS is the side opposite 60°, so it’s the side: RS = 4. The area of triangle QRS is 1/2(4)(4) = 8.
Central Angles and Inscribed Angles
An angle whose vertex is the center of the circle is called a central angle.
The degree of the circle (the slice of pie) cut by a central angle is equal to the measure of the angle. If a central angle is 25º, then it cuts a 25º arc in the circle.
An inscribed angle is an angle formed by two chords originating from a single point.
An inscribed angle will always cut out an arc in the circle that is twice the size of the degree of the inscribed angle. If an inscribed angle has a degree of 40, it will cut an arc of 80º in the circle.
If an inscribed angle and a central angle cut out the same arc in a circle, the central angle will be twice as large as the inscribed angle.
Circumference of a Circle
The circumference is the perimeter of the circle. The formula for circumference of a circle is
where r is the radius. The formula can also be written C = πd, where d is the diameter. Try to find the circumference of the circle below:
Plugging the radius into the formula, C = 2πr = 2π (3) = 6π.
Arc Length
An arc is a part of a circle’s circumference. An arc contains two endpoints and all the points on the circle between the endpoints. By picking any two points on a circle, two arcs are created: a major arc, which is by definition the longer arc, and a minor arc, the shorter one.
Since the degree of an arc is defined by the central or inscribed angle that intercepts the arc’s endpoints, you can calculate the arc length as long as you know the circle’s radius and the measure of either the central or inscribed angle.
The arc length formula is
where n is the measure of the degree of the arc, and r is the radius.
Here’s the sort of question the SAT might ask:
Circle D has radius 9. What is the length of arc AB?
In order to figure out the length of arc AB, you need to know the radius of the circle and the measure of , the inscribed angle that intercepts the endpoints of AB. The question tells you the radius of the circle, but it throws you a little curveball by not providing you with the measure of . Instead, the question puts in a triangle and tells you the measures of the other two angles in the triangle. Like we said, only a little curveball: You can easily figure out the measure of because, as you (better) know, the three angles of a triangle add up to 180º.
Since angle c is an inscribed angle, arc AB must be 120º. Now you can plug these values into the formula for arc length:
Area of a Circle
If you know the radius of a circle, you can figure out its area. The formula for area is:
where r is the radius. So when you need to find the area of a circle, your real goal is to figure out the radius.
Area of a Sector
A sector of a circle is the area enclosed by a central angle and the circle itself. It’s shaped like a slice of pizza. The shaded region in the figure below is a sector:
There are no analogies on the SAT anymore, but here’s one anyway: The area of a sector is related to the area of a circle just as the length of an arc is related to the circumference. To find the area of a sector, find what fraction of the sector makes up and multiply this fraction by the area of the circle.
where n is the measure of the central angle that forms the boundary of the sector, and r is the radius.
Try to find the area of the sector in the figure below:
The sector is bounded by a 70° central angle in a circle whose radius is 6. Using the formula, the area of the sector is
Polygons and Circles
We’ve talked already about triangles in circle problems. But all kinds of polygons have also been known to make cameos on SAT circle questions. Here’s an example:
What is the length of minor arc BE if the area of rectangle ABCD is 18?
To find the length of minor arc BE, you have to know two things: the radius of the circle and the measure of the central angle that intersects the circle at points B and E. Because ABCD is a rectangle, and rectangles only have right angles, figuring out the measure of the central angle is simple. is 90°, so the measure of the central angle is 90°.
Finding the radius of the circle is a little tougher. From the diagram, you can see that the radius is equal to the height of the rectangle. To find the height of the rectangle, you can use the fact that the area of the rectangle is 18, and the length is 6. Since A = bh, and you know the values of both a and b,
Now that you’ve got the radius and measure of the angle, plug them into the arc length formula to find the length of minor arc BE.
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