# Theorems involving chords and arcs relationship

### | CK Foundation

Chord Chord Relationships: Learn the chord chord relationships within a circle. will learn a variety of relationships within a circle that involve chords and arcs. This lesson highlights MP.7 as students study different circle relationships and and MP.3 are also highlighted as students attempt a series of proofs without . With your group, prove that if the chords are congruent, the arcs subtended by. Theorem In a circle, if two chords are equal in measure, then their Figure 2 The relationship between equality of the measures of (nondiameter) chords and equality of the Figure 5 A circle with two minor arcs equal in measure.

### Formulas for Angles in Circles - MathBitsNotebook(Geo - CCSS Math)

So they want to know the ratio of the area of the circle to the area of the square. The area of the square is just 2r times 2r.

Which is 4r squared. Area of the circle is just pi r squared. You hopefully learned the formula for area of a circle. Divide the numerator and the denominator by r squared.

CE is equal to 6. What is the value of DE. Let's call that x. Now, I'm not going to prove it here, just for saving time. But there's a neat property of chords within a circle. That if I have two chords intersecting a circle, it turns out that the two segments when you multiply them times each other, are always going to be equal to the same thing.

So in this case, 5 times So the two segments of chord AB, so 5 times That's going to be equal to these two segments multiplied by each other. It's going to be equal to x times 6. So you get 60 is equal to 6x. Divide both sides by 6, you get x is equal to And that is choice C. That might be a fun thing for you to think about after this video of why that is. And maybe you want to play around with chords and prove to yourself that that's always the case. At least that it makes intuition for you, makes sense.

RB is tangent to a circle. Tangent means that it just touches the outside of the circle right there at only one point. And it's actually perpendicular to the radius at that point. So this is the radius of that point.

Theorems on Chords and Arcs - English

The center is at A. This is a radius. And it's tangent at point B, so it's perpendicular to the radius at that point. BD is a diameter, OK, fair enough. Well A is the center, so that's kind of obvious.

So they want to know what is the measure of angle CBR. So they want to know what this angle is equal to. Well, I kind of did it inadvertently. We know that when a line is tangent to a circle, it's perpendicular to the radius at that point. So this whole angle is 90 degrees.

## Inscribed angle theorem proof

So the angle that we're trying to figure out, let's call that x. That's the complement to Subtract 25 from both sides. So that is our inscribed angle. I'll denote it by psi -- I'll use the psi for inscribed angle and angles in this video. So I just used a lot a fancy words, but I think you'll get what I'm saying.

So this is si. It is an inscribed angle. It sits, its vertex sits on the circumference. And if you draw out the two rays that come out from this angle or the two chords that define this angle, it intersects the circle at the other end. And if you look at the part of the circumference of the circle that's inside of it, that is the arc that is subtended by si. It's all very fancy words, but I think the idea is pretty straightforward. This right here is the arc subtended by si, where psi is that inscribed angle right over there, the vertex sitting on the circumference.

Now, a central angle is an angle where the vertex is sitting at the center of the circle. So let's say that this right here -- I'll try to eyeball it -- that right there is the center of the circle.

### Circle Geometry

So let me draw a central angle that subtends this same arc. So that looks like a central angle subtending that same arc. Let's call this theta. So this angle is si, this angle right here is theta. So if I were to tell you that si is equal to, I don't know, 25 degrees, then you would immediately know that theta must be equal to 50 degrees. Or if I told you that theta was 80 degrees, then you would immediately know that si was 40 degrees.

So let's actually prove this. So let me clear this.

So a good place to start, or the place I'm going to start, is a special case. I'm going to draw a inscribed angle, but one of the chords that define it is going to be the diameter of the circle.

So this isn't going to be the general case, this is going to be a special case. So let me see, this is the center right here of my circle.

I'm trying to eyeball it. Center looks like that. So let me draw a diameter. So the diameter looks like that. Then let me define my inscribed angle. This diameter is one side of it. And then the other side maybe is just like that. So let me call this right here si. If that's si, this length right here is a radius -- that's our radius of our circle. Then this length right here is also going to be the radius of our circle going from the center to the circumference.

Your circumference is defined by all of the points that are exactly a radius away from the center. So that's also a radius. Now, this triangle right here is an isosceles triangle. It has two sides that are equal. Two sides that are definitely equal.

We know that when we have two sides being equal, their base angles are also equal. Let be the line through T perpendicular to the radius OT. Let P be any other point onand join the interval OP. Hence P lies outside the circle, and not on it. This proves that the line is a tangent, because it meets the circle only at T. It also proves that every point onexcept for T, lies outside the circle. It remains to prove part b, that there is no other tangent to the circle at T.

Let t be a tangent at T, and suppose, by way of contradiction, that t were not perpendicular to OT.

Hence U also lies on the circle, contradicting the fact that t is a tangent. Tangents from an external point have equal length It is also a simple consequence of the radius-and-tangent theorem that the two tangents PT and PU have equal length. Tangents to a circle from an external point have equal length.