Difference between revisions of "Circle"
m (category) 
(expand) 

Line 1:  Line 1:  
+  A '''circle''' is a geometric figure commonly used in Euclidean [[geometry]].  
+  {{asy image<asy>draw(unitcircle,blue);</asy>rightA basic circle.}}  
== Traditional Definition ==  == Traditional Definition ==  
−  A  +  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]]. 
−  +  [[Image:circle1.PNGthumbrightThe 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{(xh)^2 + (yk)^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{(xh)^2 + (yk)^2} = r </math> which is more commonly written as  
−  <center><math> (xh)^2 + (yk)^2 = r^2  +  <center><math> (xh)^2 + (yk)^2 = r^2 </math></center> 
'''Example:''' The equation <math> (x3)^2 + (y+6)^2 = 25 </math> represents the circle with center <math> (3,6) </math> and radius 5 units.  '''Example:''' The equation <math> (x3)^2 + (y+6)^2 = 25 </math> represents the circle with center <math> (3,6) </math> and radius 5 units.  
Line 29:  Line 31:  
'''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.  
−  +  {{incompleteproof}}  
−  ==  +  ==Related Formulae== 
−  *  +  * The [[area]] of a circle with radius <math>r</math> is <math>\pi r^2</math> 
−  *  +  * The [[circumference]] of a circle with radius <math>r</math> is <math>2\pi r</math> 
−  ==Other Properties==  +  ==Other Properties and Definitions== 
+  {{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>rightA circle with a tangent and a chord marked.}}  
+  *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.  
+  *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===  
+  *[[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>?  
−  +  <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>  
−  +  ([[2006 AMC 12A Problems/Problem 16Source]])  
−  *  +  *Let 
+  
+  <math>S_1=\{(x,y)\log_{10}(1+x^2+y^2)\le 1+\log_{10}(x+y)\}</math>  
+  
+  and  
+  
+  <math>S_2=\{(x,y)\log_{10}(2+x^2+y^2)\le 2+\log_{10}(x+y)\}</math>.  
+  
+  What is the ratio of the area of <math>S_2</math> to the area of <math>S_1</math>?  
+  
+  <math> \mathrm{(A) \ } 98\qquad \mathrm{(B) \ } 99\qquad \mathrm{(C) \ } 100\qquad \mathrm{(D) \ } 101\qquad \mathrm{(E) \ } 102</math>  
+  
+  ([[2006 AMC 12A Problems/Problem 21Source]])  
+  ===Olympiad===  
== See Also ==  == See Also ==  
−  
* [[Geometry]]  * [[Geometry]]  
* [[Pi]]  * [[Pi]]  
* [[Power of a point]]  * [[Power of a point]]  
−  
* [[Homothecy]]  * [[Homothecy]]  
+  
+  [[Category:Definition]]  
[[Category:Geometry]]  [[Category:Geometry]]  
− 
Revision as of 21:59, 14 November 2007
A circle is a geometric figure commonly used in Euclidean geometry.

A basic circle. 
Contents
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.
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, , and center . We know that each point, , on the circle which we want to identify is a distance from . Using the distance formula, this gives which is more commonly written as
Example: The equation represents the circle with center and radius 5 units.
Area of a Circle
The area of a circle is where is the mathematical constant pi and 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:
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 and width thus making its area .
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 and circumference had an area equivalent to the area of a right triangle with base and height . First let the area of the circle be and the area of the triangle be . We have three cases then.
Case 1: The circle's area is greater than the triangle's area.
Related Formulae
 The area of a circle with radius is
 The circumference of a circle with radius is
Other Properties and Definitions

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
 Circles with centers and have radii 3 and 8, respectively. A common internal tangent intersects the circles at and , respectively. Lines and intersect at , and . What is ?
(Source)
 Let
and
.
What is the ratio of the area of to the area of ?
(Source)