Mathematics: Geometry - Circle Properties

Arcs
An arc is a portion of a circle. Specifically, an arc is the set of points between and including two points on a circle. An arc does not contain any points inside the circle. When a segment is drawn from the endpoints of an arc to the center of the circle, a sector is formed. A minor arc is an arc that has a measure less than 180°. A major arc is an arc that has a measure of at least 180°. Every minor arc has a corresponding major arc that can be found by subtracting the measure of the minor arc from 360°. A semicircle is an arc whose endpoints are the endpoints of the diameter of a circle. A semicircle is exactly half of a circle.

Arc length is the length of that portion of the circumference between two points on the circle.
The formula for arc length is
is the arc length, <i>r</i> is the length of the radius, and <br><img data-cke-saved-src=" />
is the angular measure of the arc in degrees, or
, where

is the angular measure of the arc in radians ().


Angles of Circles
A central angle is an angle whose vertex is the center of a circle and whose legs intercept an arc of the circle. The measure of a central angle is equal to the measure of the minor arc it intercepts. An inscribed angle is an angle whose vertex lies on a circle and whose legs contain chords of that circle. The portion of the circle intercepted by the legs of the angle is called the intercepted arc. The measure of the intercepted arc is exactly twice the measure of the inscribed angle. In the following diagram, angle ABC is an inscribed angle.


Any angle inscribed in a semicircle is a right angle. The intercepted arc is 180°, making the inscribed angle half that, or 90°. In the diagram below, angle ABC is inscribed in semicircle ABC, making angle ABC equal to 90°.


Secants, Chords, and Tangents
A secant is a line that intersects a circle in two points. The segment of a secant line that is contained within the circle is called a chord. Two secants may intersect inside the circle, on the circle, or outside the circle. When the two secants intersect on the circle, an inscribed angle is formed. When two secants intersect inside a circle, the measure of each of two vertical angles is equal to half the sum of the two intercepted arcs.

Consider the following diagram where

and
.

When two secants intersect outside a circle, the measure of the angle formed is equal to half the difference of the two arcs that lie between the two secants.

In the diagram below,
.

A tangent is a line in the same plane as a circle that touches the circle in exactly one point. The point at which a tangent touches a circle is called the point of tangency. While a line segment can be tangent to a circle as part of a line that is tangent, it is improper to say a tangent can be simply a line segment that touches the circle in exactly one point.

In the diagram below,

is a secant and contains chord

and

is tangent to circle A. Notice that

is not tangent to the circle.

is a line segment that touches the circle in exactly one point, but if the segment were extended, it would touch the circle in a second point. In the diagram below, point B is the point of tangency.


Sectors
A sector is the portion of a circle formed by two radii and their intercepted arc. While the arc length is exclusively the points that are also on the circumference of the circle, the sector is the entire area bounded by the arc and the two radii.

The area of a sector of a circle is found by the formula,
is the area, <br><img data-cke-saved-src=" />
is the measure of the central angle in radians, and r is the radius. To find the area with the central angle in degrees, use the formula,

, where

is the measure of the central angle and r is the radius.
 

P1. Given that
and
, determine the following values based on the figure:

(a)

(b)


P2. Given that
,

is tangent to the circle at B, and
, determine

(a) The angle made between

and a line tangent to the circle at A.
(b) The area of the sector of the circle between C and B.
 

P1. (a). Recall that when two secants intersect inside of a circle, the measure of each of two vertical angles is equal to half the sum of the two intercepted arcs. Also, since

and

are supplementary, the measure of
. In other words:

(b) Note that the whole circle is divided into four arcs. Thus,




P2. (a) A line tangent to the circle at A creates a right triangle with one vertex at O, one at A, and the final vertex where

intersects the tangent line, let us call that point G.

Since AB is a diameter, the line tangent at A is perpendicular to AB, so
.
The triangle COB has two legs that are the radius of the circle and so must be isosceles. So,
, which means that

and the vertical angle

both equal
.
Knowing this we can find
:



We know

and that triangle COB is isosceles with two legs equal to the radius, so a perpendicular bisector of the triangle as shown will create a right triangle:


Recall that the cosine of an angle in a right triangle is the ratio of the adjacent side to the hypotenuse. Thus, we can find r:


A. noted in part (a),
, so the area of the sector is: