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### Segments in a Circle

- Be familiar with the terms circle, chord, diameter, radius, secant, and tangent
- If a circle has circumference of C units and a radius of r units, then

### Segments in a Circle

name the circle.

Find a chord and a diameter in Circle A.

Diameter:―DE

Find a secant and a tangent of Circle A.

Tangent: ↔CD

- r = [d/2] = [14/2] = 7

- C = 2πr
- r = [C/(2π)] = [20/2*3.14] = 3.18

The radius of the circle is 5, BC = 6, find CD.

- BD = 2r = 2*5 = 10

^{2}− BC

^{2}} = 8.

Name all the chords and diameter in Circle A.

Diameter: ―BD

The circumference of Circle A is 20, BC = 4, find CD.

- r = [C/(2π)] = [20/2*3.14] = 3.18
- BD = 2r = 2*3.18 = 6.36
- CD = √{BD
^{2}− BC^{2}} = √{6.36^{2}− 4^{2}} = 4.94

Square BCDE, circle A, BC = 12, find the circumference of circle A.

- d = BC = 12
- r = [1/2]d = 6
- C = 2πr = 2*3.14*6 = 37.68.

Write all the congruent segments.

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

Answer

### Segments in a Circle

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

- Intro 0:00
- Segments in a Circle 0:10
- Circle
- Chord
- Diameter
- Radius
- Secant
- Tangent
- Circumference 3:56
- Introduction to Circumference
- Example: Find the Circumference of the Circle
- Circumference 6:40
- Example: Find the Circumference of the Circle
- Extra Example 1: Use the Circle to Answer the Following 9:10
- Extra Example 2: Find the Missing Measure 12:53
- Extra Example 3: Given the Circumference, Find the Perimeter of the Triangle 15:51
- Extra Example 4: Find the Circumference of Each Circle 19:24

### Geometry Online Course

### Transcription: Segments in a Circle

*Welcome back to Educator.com.*0000

*For this next unit, we are going to go over circles, and this first section of the unit is parts of the circle.*0003

*First, let's go over the different segments in a circle.*0012

*Now, here, right in the middle, that point right there is called the center--that is the center of the circle.*0017

*Now, whatever that point is...this point is labeled P; that means that this whole circle is going to be called circle P.*0027

*You are going to name the circle based on that center point.*0035

*And when you write it, it would be circle P; and you can also write it like this--a circle with a dot in the middle, P.*0040

*And that would mean circle P, just like we have triangle ABC and so on; this would be circle P.*0053

*Now, this first one right here: when you have a segment whose endpoints are on the circle, it is called a chord.*0061

*This right here is called a chord, and the endpoints must be on the circle.*0074

*Any time you have a segment with both endpoints on the circle, it is called a chord.*0086

*Now, this segment right here also has endpoints on the circle, so this segment is also known as a chord.*0093

*But more specifically, since it is passing through the center, this is called (I am sure you know) the diameter.*0102

*A diameter is still a chord, because any time you have a segment with endpoints on the circle, it is a chord.*0113

*But since this one is a little special, because it is passing through the center, we call it a diameter.*0120

*Now, what do we know as half the diameter? This right there is called the radius.*0127

*And then, when you have a line, this doesn't have endpoints; this is continuously going, and it is just passing through the circle at two points.*0137

*So, it is not a chord; now, if I just name this from this point to this point, then it could be called a chord, because I am calling it by its endpoints.*0149

*But this line itself, when it is just passing through the circle at two points--this is called a secant.*0160

*And again, it has to be passing through; it is different from a chord--a chord stops on one point of the circle and another point of the circle.*0174

*This doesn't stop; it just passes through, and it has to be passing through at two points--this part right here and this part right here.*0183

*Now, when you have either a line segment or a line passing through, but touching the circle in one point--*0192

*so it is like the secant; this is called a tangent, when it is touching at one point.*0202

*It could be a line segment or a line, but it just has to be touching the circle at one point; that is called a tangent.*0212

*These are the segments in a circle: chord, diameter, secant, and tangent--and radius, of course.*0220

*Next, we are going to go over circumference: if a circle has a circumference of C*0239

*(so then, C stands for circumference) and a radius of r units*0244

*(that means the r stands for radius), then C = 2πr, 2 times π times the radius; that equals circumference.*0248

*Now, what is circumference? If you take the circle right here from this point, and we are going to measure all the way around here, that is circumference.*0256

*Now, if I have a square, and I do the same thing--I start here, and I just measure all around the square,*0271

*or the rectangle, that would be called the perimeter.*0282

*A circumference and a perimeter are the same exact thing, except that perimeter is for anything with sides, and circumference is for the circle.*0289

*So, it is the perimeter of a circle; but we call it circumference, because there are no sides.*0301

*When you are given the radius--let's say that the radius is 5--then you are going to take the circumference,*0310

*and make it equal to 2 times the radius of 5, times π.*0317

*So then, the circumference would be 10π; and π, we know, is 3.14.*0323

*Well, it is more than that; it is actually an irrational number, so it goes on forever; but you can just use 3.14.*0332

*10 times 3.14 would be 31.4; that would be the circumference.*0342

*Now, you can also use a slightly different formula.*0348

*We know that, since 2 times π times r are all multiplied together, 2 times r is the diameter.*0354

*So, if I have 2, a radius right here to the center, and then from the center all the way to here, that is called the diameter.*0362

*So, you are going to multiply 2 times this number anyway, and that would be the diameter.*0370

*We can also say...instead of writing the 2 and the r, we can write d for diameter, and then π.*0376

*Circumference equals 2πr, or circumference equals diameter times π, dπ or πd.*0387

*Here, we have a right triangle within the circle, and we want to find the circumference of the circle.*0403

*To find the circumference, we need either (since this is the formula) 2πr, or the 2 and the r make up d, so dπ.*0412

*We need either the radius or the diameter.*0423

*Now, from this right triangle, the hypotenuse is the diameter of the circle.*0427

*So, as long as I can find the hypotenuse, then that would be my d.*0433

*How would you find the third side of this triangle?*0440

*Well, since it is a right triangle, I can use the Pythagorean theorem.*0443

*c ^{2} (that is the hypotenuse) = 12^{2} + 5^{2}; c^{2} = 144 + 25.*0450

*So, c ^{2} = 169, which makes c 13; so this hypotenuse has a measure of 13.*0468

*And again, since this whole thing is the diameter, and the hypotenuse and the diameter are the same thing,*0479

*I can make that d; so this is also equal to diameter.*0487

*And instead of having to divide that by 2, and then having to multiply it again by 2 and multiply by π, remember: I can just use this.*0494

*Diameter is 13 times π; remember: for π, you can just use 3.14, even though it is an irrational number (it goes on forever).*0505

*13 times 3.14 would be 40.84; and you can just use your calculator for that.*0518

*You would just do 13 times...and your calculator might actually have a button that says π;*0528

*so instead of punching in 3.14, you can push the π button; so times π, equals, and then 40.84 is your answer.*0535

*We are just going to go through our examples now.*0552

*Use a circle to answer the following: now, this circle is labeled circle C.*0556

*Now, don't get this confused with circumference; it is not the same C--this is just the name of the point, C,*0566

*and circumference is capital C; that stands for circumference.*0573

*When I say circle C, it is just the name of the circle.*0577

*Name a chord of the circle: now, I know that a chord is when a segment has endpoints on the circle.*0584

*I can say that AB is a chord, because the endpoints are right here; and DE is also a chord,*0596

*even though DE is a diameter; as long as it has two endpoints on the circle, it is a chord.*0603

*So, it is a diameter, but it is also known as a chord; it is just passing through the center.*0611

*I am going to say AB, and I can say DE.*0616

*Now, we know that CF is not a chord, because we only have one endpoint on the circle; the other endpoint is at the center, so this is not a chord.*0624

*If DE right here, the diameter, is 10, what is CF?*0635

*CF is not a chord; it is a radius; CF is a radius (a radius, we know, is when we have one endpoint on the circle, and then the other at the center).*0642

*It is a radius; and all of the radii of a circle (plural is radii) are congruent.*0655

*So, if CF has a measure of 6, then this radius right here, CG--that also has a measure of 6.*0668

*If DE is 10, then I know that CD, which is half the diameter, which is a radius, is 5.*0678

*That means that CF, since it is also a radius, would also be 5; so CF = 5.*0686

*Is CE a chord? We know that a chord has two endpoints on the circle; this one does not, so this would be "no."*0695

*Why?--because CE has only one endpoint on the circle, and the other at the center, which would make it a radius.*0708

*I am going to write that over: CE has an endpoint right there on the circle, and another on the center; so that would make that a radius, not a chord.*0755

*The next example: Find the missing measures--so they are going to give us either the radius,*0773

*the diameter, or the circumference, and then we have to find the other two.*0778

*If the radius is 6, we know that the diameter is 2 times the radius; so if the radius is 6, then the diameter is 12.*0782

*And then, to find the circumference: the circumference is the diameter times π, so you can just punch in 12 times π, and that would be 37.7.*0794

*Then, the second one: the circumference is 43.96; so to find the radius and the diameter,*0814

*I am going to take the formula, and I am going to plug in the circumference.*0822

*So, 43.96 = dπ; now, I know that π is 3.14, but again, you can just use this on your calculator.*0829

*And since I am solving for d (this is my variable), I have to get rid of that π by dividing.*0841

*I can divide the π, and then your answer will be 14, so the diameter is 14.*0850

*Then you just divide that by 2 again, and then you are going to get the radius, which is 7.*0863

*Now, the diameter for this one is 2x; we know that the diameter is 2 times the length of the radius.*0871

*So, to find the radius, I just have to take the diameter and divide it by 2.*0879

*So, if you take 2x, and you are going to divide by 2, you are going to get x.*0886

*And then, to find the circumference, the circumference is going to be dπ; it is going to be 2x times π.*0896

*And you can either leave it like that, or you can just multiply the 2 and the π together: that will be 2 times x times 3.14, which would make that 6.28x.*0909

*And that would just be your answer; when you solve for C, that is your answer, 6.28x.*0935

*Or it could be just 2x times π.*0945

*The next example: the circumference is 37.68; find the perimeter of the triangle.*0953

*We have a triangle that is inscribed; that just means that it is inside the circle, with all of the endpoints on the circle.*0960

*And it is a right triangle; they give us the circumference, and we want to find the perimeter of the triangle.*0972

*Since it is a triangle, and it has sides, we call it "perimeter" instead of "circumference."*0982

*The circumference...we are going to use π times d, because that would be the hypotenuse of this triangle.*0986

*And then, we are going to plug this in the formula, so that we can solve for d; and that will give us the hypotenuse.*0997

*37.68 = πd; then I can divide the π, and d is going to be 12.*1007

*I have a calculator here on my screen; you are just going to take 37.68, and divide it by...*1023

*and there is a π button here, so...it is going to be 12.*1032

*Then, I am going to label that onto this triangle; that is 12.*1041

*I have the hypotenuse, and I have one side, 8.*1048

*In order for me to find the perimeter, I need all of the sides--I need the measures of all three sides.*1051

*So, here, this side is missing; how would I find that side?*1056

*If it is a right triangle, and you have the measures of two sides, then you would use the Pythagorean theorem.*1061

*12 ^{2}, the hypotenuse squared, equals 8^{2} + x^{2}.*1072

*144 = 64 + x ^{2}; subtract 64; you are going to get 80 = x^{2}.*1080

*From there, take the square root; you are going to get 8.9, so this right here is 8.9.*1098

*And now, to find the perimeter, because that is what they are asking for: the perimeter of the triangle would be 12 + 8 + 8.9.*1117

*This is 20 + 8.9; that will be 28.9, so that is your answer.*1128

*Now, if you were to have units here--if this were, let's say, inches--then I know that my perimeter is just going to be in inches.*1139

*And so, also, for circumference, it is just the units; it is not units squared.*1151

*We are not looking for the area of anything--circumference and area; it is just that we keep the same units.*1157

*For the fourth example, we are going to find the circumference of each circle.*1169

*Now, in this one, we have a square, and we have an inscribed circle.*1174

*And "inscribed," again, means when you have something--a shape--inside of another shape, and all of the endpoints are touching.*1180

*Here, this is all touching here; this is circle P; the side of the square is 10; and I have to find the circumference.*1191

*Now, I know that for circumference, I need either radius or the diameter.*1207

*Let's say diameter: now, if I draw the diameter like this, I know that this diameter is going to be the same measure as this side,*1214

*because if this is the endpoint from here to here, that is the same thing as this endpoint from here to here.*1227

*And this is 10, so then, this diameter would also be 10; so the circumference is going to be 10 times 3.14, which means it is going to be 31.4.*1232

*And then, the next one: we have an inscribed triangle; endpoints are all on the circle.*1252

*This is an isosceles triangle; if this is 4, then this also has to be 4.*1264

*So again, to find the circumference, I need the diameter, or I need the radius.*1270

*The diameter is also the hypotenuse; so I am going to use the Pythagorean theorem to look for this measure right there.*1276

*4...let's see, c ^{2} = 4^{2} + 4^{2}; c^{2} = 16 + 16, or 32...√32 = 5.66.*1284

*And instead of labeling that c, I will actually label that as d, because it is the hypotenuse, but it is also the diameter.*1318

*Its diameter is that; and then, the circumference is going to be, again, π times d; so that would be 5.66 times π, or 3.14.*1330

*And your circumference is 17.77.*1350

*That is it for this lesson; thank you for watching Educator.com.*1360

0 answers

Post by Emily Engle on July 15, 2013

I see my mistake. It is a 48.16 deg. triangle. The triangle only looks like an isosceles triangle. Thank you for patience and your approach to geometry. Your lectures have been very helpful.

0 answers

Post by Emily Engle on July 14, 2013

I think there might be a problem with the numbers of extra example 3. I think that the circumference should be 35.54. A 45 deg. triangle should have equal legs.