Circles Name - Amazon S3€¦ · Equations of Circles Notes Consider the graph x 2+y 2=16. Give...

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Circles Name: Notes Date: Gustave is trying to determine where to launch his satellite so that it will travel the longest possible distance before it has to re-enter earth’s atmosphere. He has four potential orbits which each have a different amount of predicted risk of the satellite surviving passing through Earth’s atmosphere. Gustave must make a decision. Write the definition of the term and include an image or example that represents it. Term Definition Example Circle Radius Diameter Chord Tangent Circumference

Transcript of Circles Name - Amazon S3€¦ · Equations of Circles Notes Consider the graph x 2+y 2=16. Give...

 

 

Circles  Name:   

Notes  Date:   

 

 

Gustave is trying to determine where to launch his satellite so that                       

it will travel the longest possible distance before it has to re-enter                       

earth’s atmosphere. He has four potential orbits which each have a                     

different amount of predicted risk of the satellite surviving passing                   

through Earth’s atmosphere. Gustave must make a decision. 

 

 

Write the definition of the term and include an image or example that represents it. 

Term  Definition  Example 

Circle   

 

Radius   

 

Diameter   

 

Chord   

 

Tangent   

 

Circumference   

 

  

Term  Definition  Example 

Central Angle   

 

Sector   

Arc,  

Minor Arc, 

Major Arc 

 

Inscribed 

Angle  

 

 

There are a lot of terms related to circles. Which relate to distances and which relate to angles? 

Terms that refer to distances  Terms that refer to angles 

   

 

 

The Circle as a Shape  The Conceptualizer! 

The circle is a basic shape that you learned                 

about it in elementary school. Now, you are               

going to get to know it on a whole different                   

level. 

 

#couldntWeJustStayFriends? 

 

 

 

© Clark Creative Education 

  

 

Characteristics of Circles  The Conceptualizer! 

All circles are similar. They are all the               

“same” shape, as all regular polygons are.             

The only difference is in their size. 

 

As the distance across the circle gets larger,               

so does the distance around the circle. 

 

That is, as the diameter gets larger, the               

circumference gets larger. #wheelgoesround 

 

 

 

History of : Circumferenceπ   The Conceptualizer! 

The ratio of the circumference of the circle C to its                     

diameter d is , an irrational number      π        

approximated by 3.14. There is evidence has            π    

been discovered for 4,000 years by the Babylonians               

and Egyptians. 

 

The number is defined as . Then the    π         π = dC

     

circumference C is given by . And the          dC = π      

diameter is twice the radius r, so .πrC = 2  

 

 

 

History of : Areaπ   The Conceptualizer! 

Imagine the circle chopped into many narrow             

sectors (“pizza slices”). Arrange the slices           

alternating direction. If there is an infinite number               

of slices, the slices form a rectangle. Its height is                   

the radius r; its base is half the circumference,                 

.(2πr) r21 = π   

 

Thus, the area is . This is the area        h πr)(r) rb = ( = π 2          

of the original circle AND the formula we use today. 

 

If you know the radius of the circle, you can find its                       

circumference or area. 

 

 

© Clark Creative Education 

  

 

Circumference  Notes 

Find the circumference of a circle  

with a diameter of 8 cm. 

 

 

 

Area  Notes 

Find the area of a circle  

with a diameter of 6 cm. 

 

 

 

Central Angles  The Conceptualizer! 

There are in a circle.60°3   

If you take the angle between two radii. The                 

vertex of the angle is at the center of the                   

circle. It is a central angle.  

The angle cuts off an arc of the same                 

measure.  

A central angle of cuts off an arc of .0°8 0°8  

 

 

 

 

© Clark Creative Education 

  

 

Inscribed Angles  The Conceptualizer! 

An inscribed angle has its vertex on the               

circle. Its measure is half the central angle               

that cuts off the same arc.  

 

All inscribed angles that cut off the same               

arc are congruent. 

 

 

Inscribed Angles and Right Angles  The Conceptualizer! 

An inscribed angle that cuts off (or             

“intercepts”) a semi-circle, of , has a        80°1      

measure of half that -- so it is a right angle. 

 

 

 

 

© Clark Creative Education 

  

 

Inscribed Angles  Notes 

If the measure of angle BOC is , find the86°  

measures of arc BC and angle BDC . 

 

 

 

 

Arc Length  Description 

 

The central angle intercepts an arc of the               

same angle measure -- but what of the               

actual distance along the circle from A to               

B? That is the arc length. 

 

It is a portion of the circumference, in               

proportion to central angle and can be             

solved with this proportion: 

 

360°measure angle AOB = 2πr

measure arc length AB 

 

 

© Clark Creative Education 

  

 

Arc Length  Notes 

 

Find the arc length AB. 

 

 

 

Area of a Sector  Description 

 

The area between the two radii is called a                 

sector .   

The pie slice AOB is a portion of the area of                     

the entire circle and can be solved with this                 

proportion: 

 

360°measure angle AOB = πr2

area sector AOB 

 

 

 

© Clark Creative Education 

  

 

Area of Sector  Notes 

 

Find the area of sector ACB. 

 

 

 

Tangents of a Circle  The Conceptualizer! 

 

If a line touches a circle at only one point,                   

it is called a tangent.   

A few rules--  

(1) If a line is tangent to a circle, then it                   

is perpendicular to the radius drawn           

to the point of tangency. 

(2) The converse is true too. This           

relationship often creates right       

triangle Pythagorean Theorem     

Questions. 

(3) If two segments from the same point             

are tangent to a circle, then they are               

congruent. 

 

 

© Clark Creative Education 

  

 

Applications of Tangent  Notes 

Segment CD is a tangent to Circle A. 

What is the radius of Circle A? 

 

 

 

 

 

A different definition of a circle. 

We’ve always thought of a circle as a shape, but there is another                         

way to think about it --  

Every point on a circle is the same distance from the center.  

Then the circle is the locus of all points that are a constant                         

distance r from the center. 

 

 

© Clark Creative Education 

  

 

Equation of a Circle: Origin Story  The Conceptualizer! 

All points on a circle are the same distance                 

from the center. 

 

In this diagram, you can see 8 instances of                 

the 3:4:5 Pythagorean Triple right triangle.           

The radius is the hypotenuse of each             

triangle; it has length 5. 

 

All the other points on the circle are 5 units                   

away from , too, but their coordinates    0, )( 0          

are not integers.  

 

 

Equation of a Circle  The Conceptualizer! 

Since these points on the circle can be               

understood as right triangles, we can apply             

the Pythagorean Theorem to understand         

their coordinates and the radius. 

 

a2 + b2 = c2 

32 + 42 = 52 

 

But it’s not just points such as or              3, )( 4  

for which this equation “works”. It− ,− )( 3 4              

is true for all other points, too. 

 

That means, if is on this circle, thenx, )( y   

x2 + y2 = 52 

 

Now we have an equation for any circle: 

x2 + y2 = r2  

 

It is based on the Pythagorean Theorem and 

Distance Formula. 

 

 

 

 

 

 

© Clark Creative Education 

  

 

Equations of Circles  Notes 

Consider the graph . Give details      6x2 + y2 = 1      

about its size and location. 

 

 

 

Shifting the Circle  The Conceptualizer! 

What if you shift the center of the circle to 

?2, )( 3  

 

Now every point on the circle is 5      x, )( y            

points away from . By Distance      2, )( 3      

Formula: 

5 = √(x )− 2 2 + (y )− 3 2 

(x )− 2 2 + (y )− 3 2 = 52 

 

The circle is .5(x )− 2 2 + (y )− 3 2 = 2  

 

 

 

Writing Equations  Notes 

What is the equation for the circle with 

radius 2 and center ?− , )( 4 6  

 

© Clark Creative Education 

  

 

Sketch an example or an equation of each of the following. 

Circumference  Area of a Circle  Arc Length  Tangent 

       

 

 

 

Jose creates a pie chart of his expenses. Of the $1,800 he spends each month, rent accounts for                                   

$1,000. What is the central angle of the sector representing rent? 

 

 

 

 

 

 

A Ferris wheel was built on Navy Pier in Chicago. Its diameter is 140 feet. How many feet does the                                       

Ferris wheel rotate if one ride was three complete turns?  

 

  

 

 

 

A clock has a radius of 9 cm. The sector between the hands at 4 o’clock is painted blue. What area                                         

is covered by blue paint? 

 

 

 

 

© Clark Creative Education