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Difference between revisions of "Circle"

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A '''circle''' is a geometric figure commonly used in Euclidean [[geometry]].
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{{asy image|<asy>draw(unitcircle,blue);</asy>|right|A basic circle.}}
 
== Traditional Definition ==
 
== Traditional Definition ==
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A '''circle''' is defined as the [[set]] (or [[locus]]) of [[point]]s in a [[plane]] with an equal distance from a fixed point.  The fixed point is called the [[center]] and the distance from the center to a point on the circle is called the [[radius]].
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A circle is defined as the [[set]] (or [[locus]]) of [[point]]s in a [[plane]] with an equal distance from a fixed point.  The fixed point is called the [[center]] and the distance from the center to a point on the circle is called the [[radius]].  
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<center>[[Image:circle1.PNG]]</center>
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[[Image:circle1.PNG|thumb|right|The radius and center of a circle.]]
  
 
== Coordinate Definition ==
 
== Coordinate Definition ==
 
Using the traditional definition of a circle, we can find the general form of the equation of a circle on the [[coordinate plane]] given its radius, <math> r </math>, and center <math> (h,k) </math>.  We know that each point, <math> (x,y) </math>, on the circle which we want to identify is a distance <math> r </math> from <math> (h,k) </math>.  Using the distance formula, this gives <math> \sqrt{(x-h)^2 + (y-k)^2} = r </math> which is more commonly written as
 
Using the traditional definition of a circle, we can find the general form of the equation of a circle on the [[coordinate plane]] given its radius, <math> r </math>, and center <math> (h,k) </math>.  We know that each point, <math> (x,y) </math>, on the circle which we want to identify is a distance <math> r </math> from <math> (h,k) </math>.  Using the distance formula, this gives <math> \sqrt{(x-h)^2 + (y-k)^2} = r </math> which is more commonly written as
  
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<center><math> (x-h)^2 + (y-k)^2 = r^2. </math></center>
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<center><math> (x-h)^2 + (y-k)^2 = r^2 </math></center>
  
 
'''Example:''' The equation <math> (x-3)^2 + (y+6)^2 = 25 </math> represents the circle with center <math> (3,-6) </math> and radius 5 units.
 
'''Example:''' The equation <math> (x-3)^2 + (y+6)^2 = 25 </math> represents the circle with center <math> (3,-6) </math> and radius 5 units.
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'''Case 1:''' The circle's area is greater than the triangle's area.
 
'''Case 1:''' The circle's area is greater than the triangle's area.
  
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''This proof needs to be finished.''
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{{incomplete|proof}}
  
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==Formulas==
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==Related Formulae==
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* '''Area:''' <math>\pi r^2</math>
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* The [[area]] of a circle with radius <math>r</math> is <math>\pi r^2</math>
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* '''Circumference:''' <math>2\pi r</math>
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* The [[circumference]] of a circle with radius <math>r</math> is <math>2\pi r</math>
  
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==Other Properties==
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==Other Properties and Definitions==
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{{asy image|<asy>draw(unitcircle);draw((-0.8,1)--(1,1),Arrow);draw((1,1)--(-0.8,1),Arrow);draw((0,1)--(1,0));</asy>|right|A circle with a tangent and a chord marked.}}
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*A line that touches a circle at only one point is called the [[Tangent (Geometry)|tangent]] of that circle. Note that any point on a circle can have only one tangent.
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*A line segment that has endpoints on the circle is called the chord of the circle. If the chord is extended to a line, that line is called a secant of the circle.
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*Chords, secants, and tangents have the following properties:
 +
**The perpendicular bisector of a chord is always a diameter of the circle.
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**The perpendicular line through the tangent where it touches the circle is a diameter of the circle.
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**The [[Power of a point]] theorem.
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Other interesting properties are:
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*A right triangle inscribed in a circle has a hypotenuse that is a diameter of the circle.
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*Any angle formed by the two endpoints of a diameter of the circle and a third distinct point on the circle as the vertex is a right angle.
  
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* awaiting diagrams to add stuff on inscribed angles + tangents.
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==Problems==
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===Introductory===
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===Intermediate===
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*[[Circle]]s with [[center]]s <math>A</math> and <math>B</math> have [[radius |radii]] 3 and 8, respectively. A [[common internal tangent line | common internal tangent]] [[intersect]]s the circles at <math>C</math> and <math>D</math>, respectively. [[Line]]s <math>AB</math> and <math>CD</math> intersect at <math>E</math>, and <math>AE=5</math>. What is <math>CD</math>?
  
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==Practice Problems==
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<math>\mathrm{(A) \ } 13\qquad\mathrm{(B) \ } \frac{44}{3}\qquad\mathrm{(C) \ } \sqrt{221}\qquad\mathrm{(D) \ } \sqrt{255}\qquad\mathrm{(E) \ } \frac{55}{3}\qquad</math>
  
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*[http://www.artofproblemsolving.com/Forum/viewtopic.php?p=349797#p349797 2005 AMC 12A #16]
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([[2006 AMC 12A Problems/Problem 16|Source]])
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*[http://www.artofproblemsolving.com/Forum/viewtopic.php?p=423848#p423848 2006 AMC 12A #21]
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*Let
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<math>S_1=\{(x,y)|\log_{10}(1+x^2+y^2)\le 1+\log_{10}(x+y)\}</math>
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and
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<math>S_2=\{(x,y)|\log_{10}(2+x^2+y^2)\le 2+\log_{10}(x+y)\}</math>.
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What is the ratio of the area of <math>S_2</math> to the area of <math>S_1</math>?
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<math> \mathrm{(A) \ } 98\qquad \mathrm{(B) \ } 99\qquad \mathrm{(C) \ } 100\qquad \mathrm{(D) \ } 101\qquad \mathrm{(E) \ }  102</math>
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 +
([[2006 AMC 12A Problems/Problem 21|Source]])
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===Olympiad===
  
 
== See Also ==
 
== See Also ==
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* [[Dandelin Sphere]]s
 
 
* [[Geometry]]
 
* [[Geometry]]
 
* [[Pi]]
 
* [[Pi]]
 
* [[Power of a point]]
 
* [[Power of a point]]
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* [[Inversion]]
 
 
* [[Homothecy]]
 
* [[Homothecy]]
  
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[[Category:Definition]]
 
[[Category:Geometry]]
 
[[Category:Geometry]]
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[[Category:Definition]]
 

Revision as of 20:59, 14 November 2007

A circle is a geometric figure commonly used in Euclidean geometry.

[asy]draw(unitcircle,blue);[/asy]

Enlarge.png
A basic circle.

Traditional Definition

A circle is defined as the set (or locus) of points in a plane with an equal distance from a fixed point. The fixed point is called the center and the distance from the center to a point on the circle is called the radius.

The radius and center of a circle.

Coordinate Definition

Using the traditional definition of a circle, we can find the general form of the equation of a circle on the coordinate plane given its radius, $r$, and center $(h,k)$. We know that each point, $(x,y)$, on the circle which we want to identify is a distance $r$ from $(h,k)$. Using the distance formula, this gives $\sqrt{(x-h)^2 + (y-k)^2} = r$ which is more commonly written as

$(x-h)^2 + (y-k)^2 = r^2$

Example: The equation $(x-3)^2 + (y+6)^2 = 25$ represents the circle with center $(3,-6)$ and radius 5 units.

Circlecoordinate1.PNG

Area of a Circle

The area of a circle is $\pi r^2$ where $\pi$ is the mathematical constant pi and $r$ is the radius.

Archimedes' Proof

We shall explore two of the Greek mathematician Archimedes demonstrations of the area of a circle. The first is much more intuitive.

Archimedes envisioned cutting a circle up into many little wedges (think of slices of pizza). Then these wedges were placed side by side as shown below:

Pizzawedges2.PNG

As these slices are made infinitely thin, the little green arcs in the diagram will become the blue line and the figure will approach the shape of a rectangle with length $r$ and width $\pi r$ thus making its area $\pi r^2$.

Archimedes also came up with a brilliant proof of the area of a circle by using the proof technique of reductio ad absurdum.

Archimedes' actual claim was that a circle with radius $r$ and circumference $C$ had an area equivalent to the area of a right triangle with base $C$ and height $r$. First let the area of the circle be $A$ and the area of the triangle be $T$. We have three cases then.

Case 1: The circle's area is greater than the triangle's area.

Template:Incomplete

Related Formulae

Other Properties and Definitions

[asy]draw(unitcircle);draw((-0.8,1)--(1,1),Arrow);draw((1,1)--(-0.8,1),Arrow);draw((0,1)--(1,0));[/asy]

Enlarge.png
A circle with a tangent and a chord marked.
  • A line that touches a circle at only one point is called the tangent of that circle. Note that any point on a circle can have only one tangent.
  • A line segment that has endpoints on the circle is called the chord of the circle. If the chord is extended to a line, that line is called a secant of the circle.
  • Chords, secants, and tangents have the following properties:
    • The perpendicular bisector of a chord is always a diameter of the circle.
    • The perpendicular line through the tangent where it touches the circle is a diameter of the circle.
    • The Power of a point theorem.

Other interesting properties are:

  • A right triangle inscribed in a circle has a hypotenuse that is a diameter of the circle.
  • Any angle formed by the two endpoints of a diameter of the circle and a third distinct point on the circle as the vertex is a right angle.

Problems

Introductory

Intermediate

$\mathrm{(A) \ } 13\qquad\mathrm{(B) \ } \frac{44}{3}\qquad\mathrm{(C) \ } \sqrt{221}\qquad\mathrm{(D) \ } \sqrt{255}\qquad\mathrm{(E) \ } \frac{55}{3}\qquad$

(Source)

  • Let

$S_1=\{(x,y)|\log_{10}(1+x^2+y^2)\le 1+\log_{10}(x+y)\}$

and

$S_2=\{(x,y)|\log_{10}(2+x^2+y^2)\le 2+\log_{10}(x+y)\}$.

What is the ratio of the area of $S_2$ to the area of $S_1$?

$\mathrm{(A) \ } 98\qquad \mathrm{(B) \ } 99\qquad \mathrm{(C) \ } 100\qquad \mathrm{(D) \ } 101\qquad \mathrm{(E) \ } 102$

(Source)

Olympiad

See Also

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