VeJo23

Chapter 9. Circles .

9.1 Chords and arcs:

Definitions: Circles Circle: a circle is the set of all points in a plane that are equal distance from a said point in a plane known as the center of the circle. Radius: The distance from the center point of the circle to the outer rim. Chord: a segment who's endpoints land on the circle. Diameter: double the radius or the distance across the circle at the widest point.

Definitions: Central angle and intercepted Arc Central angle: the angle of a circle who's vertex is in the center of the circle. Interceptedarc: is an arc with endpoints that end on the angle.

Definitions: Degree measure of arcs: The degree measure of a major arc is 360 degrees minus the degree measure of its minor arc. degree measure of a an arc is the measure of its central angle. The the degree measure of a semicircle or half circle is 180 degrees

Arc Length: if r is the radius of a circle and M is the degree measure of an arc of the circle then the length, L, of the arc is given by the following. Chords and Arcs theorem: In a circle, or in congruent circles, the arc of a congruent chords are congruent.

Example: if there is a circle split in three parts in a pie fashion from the center point. section A's central angle is 125 Degrees what is section A's arc angle messure? Answer: The arc angle is the same as the central angle becouse the vertex is on the center point.

9.2 Tangents and circles:



Secants and tangents: A secant to a circle is a line that intersects the circle at 2 points. A tangent is a line in a plane that is the same as a circle that intersects the circle at exactly one point, which is known as the point of tangency.

Tangent theorem: If a line is tangent of a circle, then the line is to a radius of the circle drawn to the point of tangency.

Radius and chord Theorem: A radius that is perpendicular to a chord of a circle the Chord.

Converse of the tangent theorem: If a line is perpendicular to a radius of a circle at its endpoint on the circle, then chord to the circle.

Theorem: The perpendicular bisector of a chord passes through the center of the circle.

find the length of line AB: 5^2 - 3^2=4^2=16= AX=4 AX=XB XB=4

9.3 Inscribed Angles and arcs.



Inscribed angle: An angle whose vertex lies on a circle and whose sides are chords of the circle.

Inscribed angle theorem: The measure of an angle inscribed in a circle is equal to half the measure of the intercepted arc.

Right Angle corollary: If an angle intercepts a semicircle, then the angle is a right angle.

Arc-intercept Corollary: If two inscribed angles intercept the same arc. then they have the same measure.

9.4 Angles formed by Secants and Tangents: Theorem: If a tangent and a secant (or a chord) intersects on a circle at the point of tangency, then the measure of the angle formed is the same as the measure of its intercepted arc.

Theorem: The measure of an angle formed by two secants or chords that intersect in the interior of a circle is half the added measure of arcs divided by 2.

Theorem: The measure of an angle formed by two secants that intersect in the exterior of a circle is the twice the arcs of the measure of the intercepted arcs.

Example:



Find angle AEC: The angle is formed by two secants that are intersected inside the circle. 60+30=90 90/2=45 Measure angle aec = 45 degrees.

9.5 Segments of Tangents, secants, and Chords:

Theorem: If two segments are tangent to a circle from the same external point, then the segments are congruent.

Theorem: If two secants intersect outsides of a circle, the product of the lengths of one secant segment and its external segment equals.

Theorem: If a Secant and a tangent intersect outside a circle, then the product of the lengths of the segment and its external segment equals half.

Theorem: If two chords intersect inside a circle, then the product of the lengths of the segment of one chord equals the whole amount.

Example: if line Ab = 1.65 and line Bc = 2.23 while Db =1.48 what is the length of line Be?

Answer: 2.23/1.65=1.35 1.48*1.35=1.99 Line BE = 1.99

9.6 Circles in the coordinate plane:



In this chapter you will be figureing out how to find the radius and from that the circle from an equasion similar to: x^2+y^2=S

S= the radiuse squared and the x and y show the coordinates of the center of the circle.

Example: (x^2 - 5)+(y^2 + 3)=36 R=6 coordinate of the circle is the oposite of the said number next to x or y. so the coordinates for the center of the circle would be (5,-3)

you would use that to construct your circle.