Difference between revisions of "Circle"
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+ | A '''circle''' is a geometric figure commonly used in Euclidean [[geometry]]. | ||
+ | {{asy image|<asy>draw(unitcircle,blue);</asy>|right|A 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.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 | ||
− | <center><math> (x-h)^2 + (y-k)^2 = r^2 | + | <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. | ||
− | + | {{incomplete|proof}} | |
− | == | + | ==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>|right|A 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 16|Source]]) | |
− | * | + | *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 21|Source]]) | ||
+ | ===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 20:59, 14 November 2007
A circle is a geometric figure commonly used in Euclidean geometry.
|
A basic circle. |
Contents
[hide]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)