Geometry: Arcs

Arcs

I know that I've just thrown a lot of new terminology at you, but I'm not done. I've connected points on a circle with straight line segments. It is also possible to connect points on a circle using the curvy part of the circle. Suppose you have two points, A and B, on the circle, as shown in Figure 17.3. The points between A and B form the line segment ¯AB, and the points between A and B that lie on the circle make up the arc AB. Because the arc AB is curvy (it consists of part of the circle), it is given the abbreviation ˆAB.

Figure 17.2¯AB and ˆAB on a circle.

There's just one problem with this idea. There are actually two arcs that include the points on the circle between A and B, and you need to be able to distinguish between the two. You can do this by specifying a point in between A and B on the correct piece of the circle. So the upper arc is ˆADB, and the lower arc is ˆAEB. If A and B are endpoints of a diameter, ˆADB and ˆAEB are called semicircles. An arc of a circle is either a semicircle, part of a semicircle (called a minor arc), or more than a semicircle but less than an entire circle (called a major arc).

Because you can measure the lengths of line segments, it is natural to ask about measuring arcs. Arcs have length and curve. To measure the length of an arc, imagine straightening the arc and measuring its length with your ruler. The two arcs on the left in Figure 17.3 have the same length but not the same curvature. The arcs on the right have the same curvature but not the same length. So when you measure arc length, there are two features you need to address: actual length and curvature. In a circle (or in congruent circles), congruent arcs are arcs that match up both in length and in curvature.

Figure 17.3Left: Two arcs with the same length but different curve. Right: Two arcs with the same curve, but different length.

The curvature of an arc is measured in degrees. The measure of ˆAB will be denoted mˆAB. In order to measure an arc, I need to introduce you to a central angle. A central angle of a circle is an angle whose vertex is the center of the circle and whose sides are radii of the circle. The reason that central angles are needed when measuring the curvature of an arc ˆAB can be seen in Figure 17.4. The minor arc AB and the central angle ∠ACB are related to each other. Every arc has an associated central angle, and every central angle has an associated arc. If you are given a central angle, its intercepted arc is determined by the two points of intersection of the angle with the circle and all points of the arc in the interior of the angle.

Figure 17.4A central angle and its corresponding minor arc.

The correspondence between central angles and intercepted arcs can be used to define the curvature of an arc (also known as its degree measure). To solidify this idea, you will need a postulate.

• Postulate 17.1: Central Angle Postulate. In a circle, the degree measure of a central angle is equal to the degree measure of its intercepted arc.

This postulate enables you to determine the degree measurement of an arc by determining the measurement of its central angle. Because you have already explored angles in great detail, you have laid the foundation for understanding the degree measurement of an arc. A minor arc has a corresponding central angle whose measure is less than 180º; a semicircle has a central angle that is straight, so its degree measure equals 180º; and a major arc has a degree measure greater than 180º. Because the notion of degree measurement of an arc and central angle measurement are combined, central angle measurements greater than 180º are allowed. And if you throw in the Angle Addition Postulate (Postulate 4.2), you see that the degree measurement of the entire circle is twice the degree measurement of a semicircle. In other words, the degree measurement of a circle is 360º. Figure 17.5 shows some central angles and their intercepted arcs.

Figure 17.5Central angles and their intercepted arcs.

The last term I need to define for you is a sector of a circle. Imagine that you have ordered a slice of pizza. The two straight edges of the pizza can be viewed as radii of the circle (assuming it was cut properly), and the crusty part of the pizza is the arc of the circle. The entire pizza slice is a sector of the circle. A sector of a circle is the region bounded by two radii of the circle and the intercepted arc.

You have already seen the Segment Addition Postulate and the Angle Addition Postulate. There is also an Arc Addition Postulate, which says exactly what you would expect it to say.

• Postulate 17.2: Arc Addition Postulate. If B lies between A and C on a circle, then mˆAB + mˆBC = mˆAC.

You will use this postulate when you combine arcs of circles.