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Central angles give rise to another concept that we call the arc. Not ark, as in Raiders of the Lost Ark. More like Noah's ark…but only slightly.
It's hard to define what arcs are formally. Bow ties and ball gowns really don't do it for us, anyway. However, we can describe them fairly well like this: an arc is a curved segment of a circle. Like a line segment, every arc has two endpoints.
Notice that points A and B define two arcs at the same time—there's the arc that directly connects A and B (arc 1, in the figure), and there's the arc that connects A and B by way of point C (arc 2, in the figure).
Every arc has a buddy, just as every central angle has a buddy. To distinguish each arc from its buddy, we usually add points to our figure. In the figure above, we would call arc 2 "arc ACB."
See? Just like the animals on Noah's ark, they come in pairs.
Buddy arcs complete each other just as central angles do. They even hold hands at their endpoints. If that's too mushy-gushy, imagine them shaking hands instead, like esteemed businessmen.
For any arc with two distinct endpoints, we can draw a central angle by drawing two rays, each starting at the center of the circle and going through one of the endpoints. We say that such an angle intercepts that arc, or that the arc subtends that angle. Likewise, any central angle intercepts an arc. Below, ∠AOB intercepts arc AB.
The measure of a central angle tells us something about its intercepted arc: how "much of a circle" that arc is. The measure (sometimes called the degree measure) of an arc is equal to the measure of the central angle that intercepts that arc.
In the figure above, both ∠AOB and arc AB have measures of 90°. Both ∠COE and arc CDE have measures of 180°.
Every arc falls into exactly one of the following categories, depending on its measure.
1. An arc with measure less than 180° is a minor arc.
2. An arc with measure greater than 180° is a major arc.
3. An arc with measure equal to 180° is a semicircle.
Note that a major arc's buddy is always a minor arc, and vice versa. A semicircle's buddy is always a semicircle. They're basically twins, like Lindsay Lohan and…uh…Lindsay Lohan.
An arc has a measure of 73°. Is its buddy a minor arc, a major arc, or a semicircle arc? What is its measure?
Since 73° is less than 180°, we know that the first arc is a minor one. Since the buddy of a minor arc is always a major arc, we know right off the bat that our arc is major. We can find its measure by subtracting our original arc from the total measure of the circle. That gives us 360° – 73° = 287°. Our arc has a measure of 287°, much larger than 180° and definitely major.
In addition to its measure, arcs have another property called arc length. The length of an arc is the distance a bug would have to crawl from one endpoint of the arc to the other, while staying on the circle the whole time.
It's important to remember that arc length and arc measure are NOT the same.
If you're ever unsure whether you're dealing with arc length or arc measure, check the units. Arc measure is almost always given in units of degrees. Arc length, on the other hand, is a length, and so it may be given in units of meters, feet, inches, kilometers, or unhelpfully, "units," but certainly not degrees. (Don't be afraid of units. They're there to help you.)
It's pretty important to be able to tell when two things are different and when they're the same. It's a good skill to have, regardless of how you feel about circles, geometry, or Sesame Street.
This figure might represent you and a friend running around a circular track—you in the inside lane and your friend in the outside lane. If you both complete the same fraction of the track (say, a quarter-lap), does that mean you both run the same distance?
We have two arcs, and , intercepted by the same central angle. By definition, arcs and have the same measure. But they don't look like they have equal length, do they? We can solve this mystery, but we need more clues. Even Sherlock Holmes couldn't work with nothing.
The arc length of a circle (that is, the distance a bug would have to crawl to go around the circle exactly once, staying on the circle the whole time) is called its circumference.
The ratio of a circle's circumference to twice its radius is a constant, which we represent with the Greek letter π ("pi," pronounced "pie," like the circle-based dessert).
In symbols, this gives us the formula
C = 2πr
C is the circumference of the circle and r is its radius. The constant π is an irrational number (but nobody's perfect). We can't represent it exactly as a ratio of two integers, and if we write it in decimal form we find that it's an infinite, non-repeating decimal. We can use approximations: π is approximately equal to 3.14, or 22/7. Those approximations get the job done pretty well. However, if we want to be exact, we have to simply write π.
What is the circumference of a circle with radius r = 7 inches?
All we need in order to calculate the circumference of a circle is the radius. Well, that and the formula. Luckily, we have both. We can substitute 7 for r in the formula C = 2πr to end up with C = 2π(7) = 14π ≈ 44 inches.
Now we know how to find the circumference of a circle given its radius. We also know how to find the length of an arc with measure 360° given its radius, because that's the same thing as a circle. Two birds with one stone.
How about a semicircle? Can we find the length of a semicircle, given its radius? Spoiler alert: yes. It ought to be half the circumference of a circle with the same radius, don't you think? And let's say we wanted to find the length of a quarter-circle with a given radius. Shouldn't it be one-fourth of the circumference of a circle with the same radius?
We can generalize that idea like this: The length L of any arc with radius r and measure θ can be expressed as follows:
The above formula asks us, "How much of a circle is the arc?" and "How big is the circle?" and gives us the length of the arc. This will be a powerful tool in figuring out the Case of the Unfair Track Run.
Suppose a circular track has a radius of 50 m and the distance between the inner lane and outer lane is 10 m. If you ran a quarter-lap along the inner lane and your friend ran a quarter-lap along the outer lane, what's the length that you ran? How far did your friend run?
Since this is an arc length problem, we know we'll need to use the arc length formula. That means we need to know θ and r for both tracks. Since you and your friend both run one-fourth of a lap, that translates to θ = 360° ÷ 4 = 90°.
The radii of the tracks, on the other hand, are different. You run on the inner track, which has a radius of 50 m. Your friend runs on the outer track, which has a radius of 50 + 10 = 60 m. Plugging those values into the formula, we know that you ran about 78.5 m, while your friend ran 94.2 m.
Your friend ran a greater distance than you, even though you both ran through the same angle. In other words, you and your friend ran arcs of different lengths, even though they had the same measure. A good mathematician has the power to be a bad friend. But you can always offer your friend some pi to make up for it.
In geometry, we need to be able to prove whether two shapes are different or the same (congruent). For that, we need a formal definition of congruence for each shape we study. We've been talking about arcs and circles for a while now without a formal definition. But we'll get it out of those jeans and sneakers and into some dress shoes and a tux. How's that for formal?
If two circles have congruent radii, then they're congruent circles. If two arcs are both equal in measure and they're segments of congruent circles, then they're congruent arcs.
Notice that two arcs of equal measure that are part of the same circle are congruent arcs, since any circle is congruent to itself.
Let's circle back (pun intended) to the track example. Is the arc you travel as you run in the inner lane of the track congruent to the arc your friend travels as he runs in the outer lane? No, because the two arcs are not segments of congruent circles. They have different radii.
However, the original question asked whether you and your friend run the same distance. That question is about arc length, not arc congruence.
Congruent arcs have equal length (you can prove this yourself). Does that mean all arcs of equal length are congruent? Nope. (You can prove this yourself too.) That's like saying, "All cars can travel at 65 miles an hour, so everything that travels 65 miles an hour is a car." That's untrue, not to mention insulting to a good number of cheetahs.
We can relate central angles to arcs using the Angle-Arc Theorem: In congruent circles, two central angles are congruent if and only if their intercepted arcs are congruent. This is a biconditional statement, meaning that it goes both ways.
The first way: If two arcs are congruent, then the two central angles that intercept them are congruent. The second way: If two central angles are congruent, then the arcs they intercept are congruent.
To prove a biconditional statement, we have to prove the statement in both directions. In other words, we have to prove two statements. Let's start with the first one.
We're given that ⊙O is congruent to ⊙O' and arc AB is congruent to arc A'B'. To prove that ∠AOB is congruent to ∠A'O'B', we can say that by the definition of congruence of arc, mAB = mA'B'. By definition of arc measure, m∠AOB = m∠A'O'B'. By definition of congruence of angle, ∠AOB is congruent to ∠A'O'B'. With biconditional statements, we can't always just reverse the argument to get the reverse implication, but in this case we can.
If ∠AOB is congruent to ∠A'O'B', that tells us m∠AOB = m∠A'O'B'. By definition of arc measure, mAB = mA'B'. We're also given that ⊙O is congruent to ⊙O'. Since arcs AB and A'B' have the equal measure and are segments of congruent circles, we can say by definition of congruent arcs that arcs AB and A'B' are congruent.
That "if-and-only-if" part makes a statement much stronger because it's fortified from both ends. It's like a multivitamin for mathematical statements. Only without that horrible lodged-in-your-throat feeling.
One more thing about arcs before we move on. We can add them, just like we can add numbers. It seems silly to add shapes, doesn't it? What does that even mean?
For those deep, deep questions such as "what does arc addition mean?" we need something more than a simple definition. Enter postulates. More specifically, the Arc Addition Postulate.
Given two arcs in the same circle AB and BC with exactly one point in common (the endpoint B), we say: arc AB + arc BC = arc ABC. Of course, this also means mAB + mBC = mABC.
Arc addition will come in handy later. Trust us.
Also, don't get overambitious with postulates. They're helpful and all, but they can't answer every question for us. For instance, it's usually a bad strategy to write, "I postulate that I will get full credit on this exam." You'll probably end up with zero credit and a massive grounding.
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