SATII数学考试快递——Circles
Circles are another popular plane-geometry test topic. Unlike polygons, all circles are the same shape and vary only in size. Circles have certain basic characteristics, and test questions will focus on your understanding of these properties.
Basic Definitions of Circles
A circle is the collection of all 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), which is the most important measurement in a circle. If you know the radius of a circle, 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, making sure to pass 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
A line that intersects the circle at only one point is called a tangent line. The radius whose endpoint is the intersection point of the tangent line and the circle is always perpendicular to the tangent line.
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.
The Math IC often includes tangent lines in the test. For example:
|
If RS is tangent to circle Q, then QR is perpendicular to RS, and therefore QRS is a 30-60-90 triangle. Given that QR = 4, we know that RS = 4
, and 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 in a circle that originate 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 40º, 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 of a circle is the length of the 360º arc that forms the circle. In other words, if you were to trace around the edge of the circle, it is the distance from a point on the circle back to itself. The circumference is the perimeter of the circle. The formula for circumference is:
where r is the radius. The formula can also be written C = πd, where d is the diameter. Using the formula, try to find the circumference of the circle below:
Plugging the radius into the formula, C = 2πr = 2π (3) = 6π.