Difference between revisions of "Circle"
Winterrain01 (talk | contribs) m (→Olympiad) |
Winterrain01 (talk | contribs) (→Olympiad) |
||
Line 87: | Line 87: | ||
===Olympiad=== | ===Olympiad=== | ||
*Consider a [[circle]] <math>S</math>, and a [[point]] <math>P</math> outside it. The [[tangent line]]s from <math>P</math> meet <math>S</math> at <math>A</math> and <math>B</math>, respectively. Let <math>M</math> be the [[midpoint]] of <math>AB</math>. The [[perpendicular bisector]] of <math>AM</math> meets <math>S</math> in a point <math>C</math> lying inside the [[triangle]] <math>ABP</math>. <math>AC</math> intersects <math>PM</math> at <math>G</math>, and <math>PM</math> meets <math>S</math> in a point <math>D</math> lying outside the triangle <math>ABP</math>. If <math>BD</math> is [[parallel]] to <math>AC</math>, show that <math>G</math> is the [[centroid]] of the triangle <math>ABP</math>. | *Consider a [[circle]] <math>S</math>, and a [[point]] <math>P</math> outside it. The [[tangent line]]s from <math>P</math> meet <math>S</math> at <math>A</math> and <math>B</math>, respectively. Let <math>M</math> be the [[midpoint]] of <math>AB</math>. The [[perpendicular bisector]] of <math>AM</math> meets <math>S</math> in a point <math>C</math> lying inside the [[triangle]] <math>ABP</math>. <math>AC</math> intersects <math>PM</math> at <math>G</math>, and <math>PM</math> meets <math>S</math> in a point <math>D</math> lying outside the triangle <math>ABP</math>. If <math>BD</math> is [[parallel]] to <math>AC</math>, show that <math>G</math> is the [[centroid]] of the triangle <math>ABP</math>. | ||
− | (< | + | (<url>viewtopic.php?=217167 Source<url>) |
== See Also == | == See Also == |
Revision as of 15:24, 1 October 2020
A circle is a geometric figure commonly used in Euclidean geometry.
|
A basic circle. |
Contents
[hide]Definition
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.
Circumference and Area
Given a circle of radius , the circumference (distance around a circle) is and the area is . Both formulas involve the mathematical constant pi ().
Archimedes' Proof of Area
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.
Case 2: The triangle's area is greater than the circle's area.
Case 3: The circle's area is equal to the triangle's area.
Assume that . Let be the area of a regular polygon that is closest to the circle's area. Therefore we have so . Let the apothem be and the perimeter be so the area of a regular polygon is one half of the product of the perimeter and apothem. The perimeter is less than the circumference so and the apothem is less than the radius so . Therefore . However it cannot be both and . So .
This article is a stub. Help us out by expanding it.
Area Proof Using Calculus
Let the circle in question be , where r is the circle's radius. By symmetry, the circle's area is four times the area in the first quadrant. The area in the first quadrant can be computed using a definite integral from 0 to r of the function . Using the substitution gives the indefinite integral as , so the definite integral equals . Multiplying by four gives the area of the circle as .
Lines in Circles
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. The longest chord of the circle is the diameter; it passes through the center of the circle.
When two secants intersect on the circle, they form an inscribed angle.
Properties
- The measure of an inscribed angle is always half the measure of the central angle with the same endpoints.
- Since the diameter divides the circle into two equal parts, any angle formed by the two endpoints of a diameter and a third distinct point on the circle as the vertex is a right angle.
- Also, a right triangle inscribed in a circle has a hypotenuse that is a diameter of the circle.
- Similarly, if a tangent line and a secant line intersects at the point of tangency, the measure of the angle formed is always half the measure of the central angle with the same endpoints.
- From that property, the angle formed by the diameter and a tangent line with the point of tangency on the diameter is a right angle.
- The perpendicular line through the tangent where it touches the circle is a diameter of the circle.
- The perpendicular bisector of a chord is always a diameter of the circle.
- When two chords and intersect at point inside the circle, .
- When two chords and intersect at point outside the circle, .
- Lengths of chords can be calculated by using the Power of a point theorem.
Problems
Introductory
- Under what constraints is the circumference (in inches) of a circle greater than its area (in square inches)?
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)
Olympiad
- Consider a circle , and a point outside it. The tangent lines from meet at and , respectively. Let be the midpoint of . The perpendicular bisector of meets in a point lying inside the triangle . intersects at , and meets in a point lying outside the triangle . If is parallel to , show that is the centroid of the triangle .
(<url>viewtopic.php?=217167 Source<url>)