Back To CourseGeometry: High School
15 chapters | 160 lessons
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Jeff teaches high school English, math and other subjects. He has a master's degree in writing and literature.
Once upon a time, I wanted to be a cartoonist. I read a lot of Calvin and Hobbes, Peanuts and The Far Side, and I was pretty sure being a cartoonist was the best job ever.
I used to get those books on how to draw cartoon characters. I didn't know how to draw, but the books said to start with circles. Then you add lines to the circles and, miraculously, cartoon characters emerged. I could never quite figure out the circle thing well enough to make my own Calvin and Hobbes. In fact, the comics I drew were really quite terrible. But I did learn a thing or two about circles.
The different lines drawn on circles have different names. This one, where the line starts and ends on the circumference, is called a chord. The part of the circle it intercepts is an arc. If a line hits two points on a circle, but keeps going, it's a secant. These also form arcs. But if the arc shrinks because the two points get closer and closer and closer until the line only hits the circle at one point, we have a tangent. Somehow a combination of chords, secants and tangents on a circle can make you a cartoon character. How? I really don't know.
This is what it looks like when I try to draw a face. Yeah, not good. But you know what is good? Finding the measure of these angles.
How we do this depends on the kind of lines we're working with. These lines are chords. With two chords, we have vertical angles here and here. The chords also create arcs. And it's the relationship between the arcs and angles that matters. The measure of the angle formed by two chords is 1/2 of the sum of the intercepted arcs. Let's see this in action.
In my botched drawing, the intercepted arcs are arc AD and arc CB. We want to know the measure of angle AED. We know that arc AD is 175 degrees and arc CB is 85 degrees. So we can say that the measure of angle AED equals 1/2 the measure of arc AD plus arc CB. 175 plus 85 is 260. What's half of 260? 130. So angle AED is 130 degrees.
Now, angle CEB is going to also be 130, since it's a vertical angle. And since angle CEB and angle AEC are supplementary angles, we know angle AEC is 50 degrees, as is angle BED. With just those two arcs, we figured out four angles. Not bad! Though, again, not much of a face.
Sometimes when I draw lines on circles, I miss almost completely, like this. But this is another chance to learn about angles. Line AC is a tangent to our circle. And it forms an angle with chord AB. What's the measure of angle BAC? The measure of the angle formed by a chord and a tangent is 1/2 the measure of the arc the chord creates.
Before we get to my botched drawing, think about a tangent line and a diameter. These lines are perpendicular, so we know the angle is 90 degrees. And what's the measure of this arc? It's half the circle, so it's 180. Did we just prove our formula? I think we did.
As for my weird art, arc AB is 106 degrees. So if angle BAC equals 1/2 the measure of arc AB, then angle BAC is half of 106, which is 53 degrees. That's all there is to it!
You know what I can draw? Hats. Check out this awesome party hat. This is not just a hat, it's two tangent lines. And guess what? That means it's time for more math - party hat math.
The angle created when two tangent lines meet is half of the measure of difference in measure of the two arcs. We just subtract the minor, or smaller, arc from the major, or larger arc, then cut that in half.
In my hat, the arcs are 240 and 120 degrees. So we just do 240 minus 120, which is 120. Then we cut that in half to get 60. So angle BAC is 60 degrees. Party on, hat.
Sometimes, my hat drawing goes awry. Then it looks like this. That's some bad hat. But it's also a tangent line and a secant. And, yep, there's a formula for that. It's going to look familiar.
The angle created by a tangent and a secant is equal to half the measure of the difference in measure of the two created arcs. So when we have a tangent and a secant, we do the same basic thing as with two tangent lines. We subtract the smaller arc from the larger one, then cut it in half.
Here, we subtract the measure of arc BC from arc BD. Arc BD is 165 degrees. Arc BC is 95 degrees. 165 minus 95 is 70. Half of 70 is 35. So angle BAD is 35 degrees - and, again, a pretty bad hat.
How about one more? This hat is even worse than the last one. Instead of a tangent and a secant, we have two secants. But at least we can find the measure of the angle. It's pretty much the same as before.
The angle created by two secant lines is equal to half the measure of the difference in measure of the two created arcs. So angle CAE is just the measure of arc CE minus the measure of arc BD. Arc CE is 100 degrees. Arc BD is 40 degrees. 100 minus 40 is 60. Half of 60 is 30. So angle CAE is 30 degrees.
In summary, we didn't learn much about cartooning. But we learned a lot about angles. First, we learned that the angle formed by two chords equals half the sum of the measures of the intercepted arcs. Next, we looked at the angle formed by a chord and a tangent line. This is half the measure of the arc the chord creates.
Then, we looked at an angle formed by two tangent lines. This time, we subtracted one arc from the other, then cut that in half. With a tangent and a secant line, we also found the different in measures of the two arcs, then, again, cut that in half. Finally, we learned that angle formed by two secants is just half the difference in measures of the two intercepted arcs.
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Back To CourseGeometry: High School
15 chapters | 160 lessons