Proof: Angle between a tangent and a chord
About this video
The video proves that the angle formed by a tangent and a chord is half the measure of its intercepted arc, for both an acute and an obtuse angle. Drawing radii to make an isosceles triangle, and using the fact that a radius meets a tangent at 90 degrees, it shows the central angle (equal to the arc) is 2 theta, or that the major arc is 2 theta in the obtuse case.
You'll be able to
- state that a tangent-chord angle is half its intercepted arc.
- use a radius perpendicular to a tangent and an isosceles triangle in a proof.
- find a tangent-chord angle or arc when given the other.
Key words
- tangent
- A line that touches a circle at exactly one point.
- chord
- A segment joining two points on the circle.
- intercepted arc
- The arc that lies inside the angle.
- major arc
- An arc bigger than a semicircle; it is named with three letters, like arc BEC.
- isosceles triangle
- A triangle with two equal sides, so the angles opposite them are equal.
Transcript
- [Instructor] What we wanna do in this video is prove that the measure of the angle formed by the intersection of a tangent and a chord is half the measure of the intercepted arc. So, let's just make sense of what we're trying to prove and then we will prove it.
So, we have two situations where we have an angle formed by a tangent and a chord. So, there's one situation where it is acute, that's the scenario on the left. There's another situation where the angle formed between the chord and the tangent is obtuse. Now, what we wanna do in either of these cases is compare them to the measure of the intercepted arc.
So, in this situation right over here on the left, the intercepted arc is what I'm showing here in the yellow. So, in this left scenario, what we want to show is that theta is equal to, we wanna do, we wanna show that is half the measure of the intercepted arc, so we wanna show that it is half the measure of arc BC.
That's what we wanna show in this scenario. In this scenario, we wanna show that theta is equal to half the measure, and here we're gonna be thinking about the major arc. So, this is the intercepted arc in this scenario. So, we want to go all the way around, and for major arcs, we use three letters to show that we're going the long way around, so this is the measure of arc, we could call this BEC.
BEC. So, let's get to it. So, in this left scenario, what I'm gonna do is draw some radii. So, this over here and this over here, this is a circle, so the radius is constant, so we know that this length is equivalent to that length. We also have proved in other videos, or you might know that the angle between a radius and a tangent is going to be 90 degrees, so that right over there is 90 degrees, or another way you could think about it is this would also be 90 degrees.
Well, if we know that, if we know that this angle is 90 degrees, then what is the measure of this angle right over here? Well, that angle plus theta has to be equal to 90 degrees, so this is going to be 90 minus theta. Well, what we're dealing with here is an isosceles triangle, so if this angle is 90 minus theta, then this angle right over here, 'cause these two sides are congruent, so this is also going to be 90 minus theta.
So, my question is, what is this angle going to be? And this angle's measure is the same as the measure of arc BC. The angles in a triangle add up to 180, so this is going to be, lemme just color another color right over here, so it's gonna be 180 minus this, minus 90 minus theta, minus 90 minus theta.
That's the measure of this angle, and so this is going to be equal to, if I just distribute the negatives and simplify it, this is 180 minus negative 90, negative 90, so minus 180, and then if I multiply that times theta is plus theta plus another theta, so plus two theta, and so those cancel out so were just left with two theta.
This angle right over here is two theta. That is the measure of arc BC. So, we see that indeed, this angle is half of this angle, or this angle is half the measure of the intercepted arc. Well, now let's do the same exercise over here, and we're gonna do it in a very similar way.
Let me draw some radii. We know that this length is equal to that length, and we know that this is 90 degrees because DB is a tangent, and so now, this angle here is going to be not 90 minus theta, but theta minus this 90, so theta minus 90. Well, because this is an is isosceles triangle, this is also going to be theta minus 90, And then based on that we can figure out this angle.
Let me do it right over here. This angle is going to be 180 minus theta minus 90, theta minus 90, minus theta minus 90, theta minus 90, and so this is going to be 180, and then we're going to add 90 twice. So, plus 180, and then we're gonna subtract theta twice, so minus two theta, so it's equal to 360 minus two theta.
Well, that's interesting. If this is 360 minus two theta, what is going to be the measure of this angle right over here? This larger angle right over here, plus this angle is going to add up to 360, 'cause we're going all the way around the circle. So, it's going to be 360 minus 360 minus two theta.
I think you see where this is going. And that's gonna be 360 minus 360 plus two theta, and those cancel out and we are left with two theta. In this situation with an obtuse angle, it intersects this major arc, which once again has a measure of two theta.
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