Circle properties. In the previous lesson, we covered some of the basics. We covered radii, chords, diameter, circumference, the famous formula for circumference and the famous formula for area. Now if any of those are brand new to you, I suggest watching that previous lesson on circles. Show Transcript
In this lesson we're gonna talk about something different, in this lesson we're gonna explore several ideas involving angles in circles. The first is probably obvious, and triangle with two sides that are radii has to be isosceles. So for example, AOB has to be an isosceles triangle because OA and OB are radii. And of course, all radii to the same circle are equal.
So that would mean that the two base angles are equal. So angle B would have to be 70, and the angle in the middle would have to be 40 degrees. If the chord side of such a triangle is also equal to the radius, then the triangle would be equilateral. So in other words, if we have a chord EF that has the length equal to a radius, then when we draw that triangle, all three sides are radii.
So we get an equilateral triangle, all the angles are 60 degrees. And notice that that angle EOF, that angle is at the center of the circle. It's vertex, it's point, it's at the center of the circle. An angle with its vertex at the center of the circle is called a central angle. Now this is an important idea that test is not necessary gonna make you know that term, but this is an important idea to understand and understand how it It's related to some of the other angles.
A central angle has a unique relationship with the arc and intersects. One way to talk about the size of an arc is to talk about it's arc measure. In other words, how many degrees it takes up. The measure of the central angle equals the measure of the arc. So here, we have a central angle of 135 degrees, o that arc, JKL, that arc has to be a 135 degrees also.
This is one of the ways to talk about the size of an arc. The other way is arc length which we'll discuss on the next video. Now a diameter, essentially, is a 180 degree angle. It's a straight angle, the degrees of angle AOB would be 180. So it divides the circle into two 180 degree arcs. An arc with a measure of 180 degrees is called a semicircle.
The measure of the entire circle is 360 degrees, which is the angle all the way around the circle. If two different central angles in the same circle have the same measure, then they will intersect arcs of the same size. And so, if those two angles are equal, then the arcs have to be equal, in effect, vice versa, the arcs are equal, then the angles have to be equal also.
Similarly, equal chords in the same circle intersect arcs of equal length. So if JK = LM, then those two arcs are equal. Here's a practice problem, pause the video and then we'll talk about this. Okay, so we have that the arc AB is 50 degrees, and so we wanna know, not the angle of the center, we wanna know the angle at A, what is AOB? Well first of all, we know that the arc has to have the same measure as the central angle, so AOB equals the measure of the arc that's 50 degrees.
And we know it's an isosceles triangle, so we don't know the angle in AOB but we know it has to be equal, so we'll call that x. So 50 + x + x = 180. Well, in other words 2x = 130. x equals 130 divided by 2 which is 65. And so that's the measure of both the angle at A and the angle at B.
We want the angle at A, so that's answer choice D. Central angles have their vertices at the center of the circle. Another kind of angle has its vertex on the circle. So the vertex of this angle right here is not at the center at all, it's actually sitting on the circle itself. So this is another kind of angle, and this kind of angle is called an inscribed angle.
An inscribed angle is an angle that has its vertex on the circle itself. The side of an inscribed angle are always two chords that meet at that vertex. So here AB is a chord, and BC is a chord, and they meet at point B. An inscribed angle also has a special relationship with the arc it intercepts. Intersects, the measure of the inscribed angle is half the measure of the arc it intercepts.
So if the angle DEF is 40 degrees, then that arc, arc FGD has to be 80 degrees. Now why is this true? Here's one way to see why that's true. Suppose we a draw very special diagram like this. So, of course, arc BC equals the central angle, BOC.
We know that AOB is an isosceles triangle, so angle BAO equals angle ABO. So those two red angles are equal, just cal that x. We know that if we look at the angles forming the diameter AOC, it must be true that those two angles, the blue angel and the orange angle, have to add up to 180 degrees.
So BOC plus angle AOB have to equal 180 degrees. But it's also true in the triangle that the 2xs + AOB = 180 degrees. Well, look at those two equations. Thing + AOB = 180 degrees, thing + AOB equals 108 degrees, so in other words, those two things have to be equal. So it must be true that 2x = angle BOC, and then we can just divide, so X is half the measure of angle BOC, and so it's half the measure of the arc.
And so, this is one very special way to see why the inscribed angle has to be half the measure of the arc. This means that any inscribed angle that intercepts a semicircle, that is any inscribed angle that intercepts the endpoints of a diameter, has to be a right angle. So merely the fact that HOJ is a diameter, automatically means that K has to have a right angle, because it's intersecting an arc of 180 degrees.
And the test absolutely loves that fact, it really loves to draw this kind of diagram and expect you to know that that is a right angle. If two inscribed angles in the same circle intercept the same arc or the same chord on the same side, then those two inscribed angles have to be equal to each other. So, here those two angles LNM and LPM, those two angles are equal. Here's a practice problem, pause the video and then we'll talk about this.
Okay, so we're given angle BEC, is 20 degrees. So we know right away that that arc, arc BC, has to equal 40 degrees. But we're told that BC equals CD so we know that arc CD also equals 40 degrees. So, the arc from B to C to D, is 40 plus 40, or 80 degrees. Arc BD is 80 degrees, that's the arc that angle A intercepts.
So A has to be half that arc, or 40 degrees. Finally, we'll discuss a line outside of the circle. A tangent line is a line that passes by a Circle and touches it only at one point, tangent from the Latin tangera, to touch. And in fact, we get the English word tangible from this same Latin root. So a tangent line simply touches the circle at one point.
If we draw a radius to the point of tangency, the radius on the tangent line are always perpendicular. That's a really important fact, and the test love that fact too. Notice that this lends itself to the Pythagorean theorem, and other special right angle facts. And so there's all kinds of things that can be implied once you have a right angle.
Here's a practice problem, pause the video and then we'll talk about this. So this a complicated diagram, but it's actually not too hard of a problem. First of all, notice that angles PQS, this angle here, and this angle PRS, they intersect the same chord, the chord PS. And if two angles intersect the same core,they have to be equal.
So that means that PRS is an angle of 40 degrees. Well, now.look at the angle at S. We know that pr is a diameter. And PSR is an angle inscribed in a semicircle, so that has to be a right angle. So PSR is 90 degrees.
Well, now if we just look at the triangle PSR, we have the angle at R, we have the angle at S, we can find the angle at P. The angle the P has to be 50 degrees. Then we know that TPO is a right angle, because TP is a tangent line and it has to be perpendicular to the radius. So TPS, the angle that we're looking for, equals TPO minus SPR, which is 90 minus 50 or 40 degrees.
And we choose answer choice C. In summary, if two sides of a triangle are radii, the triangle is isosceles. A central angle has the same measure of the arc it intercepts. Equal chord lengths intercept equal arcs. An inscribed angle has half the measure of the arc it intercepts. An angle inscribed in a semicircle is 90 degrees.
Two inscribed angles intersecting the same chord on the same side are equal. And finally, a tangent line is perpendicular to a radius, at the point of tangency.