Chapter 10 - Content Notes

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10.1 - Circles and Circumference

vocabulary:

circle - the locus of all points in a plane equidistant from a given point called the center of a circle.

center - the central point where radii form a locus of points called a circle. this is how to name a circle. (ex. circle C)

radius - in a circle, any segment with endpoints that are the center of a circle and a point on the circle. all radii are congruent.

diameter - in a circle a chord that passes through the center of a circle. 2x the radius.

circumference - the distance around a circle, is measured by the equation c = 2(pi)(r).

chord - for a given circle, a segment with endpoints that are on the circle. the diameter is the longest chord in a circle, the diameter is a chord, but a chord is not always a diameter.

secant - any line that intersects a circle in exactly two points. there can be secant segments as well.

tangent - a line or segment in a plane of a circle that intersects the circle in exactly one point. circles can be tangent as well.

point of tangency - for a line that intersects a circle in only one point, the point of tangency is the point where the line and circle intersect.

common tangents - a line or segment that is tangent to two circles in the same plane.

congruent circles - congruent circles are two or more circles that have congruent radii (or diameters).

concentric circles - coplanar circles with the same center. name by: circles center + radius

arc - a part of a circle that is defined by endpoints. two points on the circle and every point in between.

minor arc - an arc with a measure less than 180 degrees, name with two letters.

semicircle - an arc that measures 180 degrees. central angle is the diameter. name using three letters.

major arc - an arc with a measure greater than 180 degrees. name with three letters.

central angle - an angle that intersects the circle in two points and has its vertex at the center of the circle.

arc measure - arc measure is the measure of an angle that the arc creates in the center of the circle. not the same as arc length.

congruent arcs - arcs in the same circle or in congruent circles that have the same measure.

intercepted arcs - an angle intercepts an arc if and only if the endpoints on the arc lie on the angle. all points of the arc except the endpoints are in the interior of the circle. each side of the angle has an endpoint on the arc. intercepted arcs are two times the measure of the intscribed angle.

inscribed - a polygon is inscribed in a circle if each of its vertices lie on the circle. inscribed is in relation to the polygon (or circle).

inscribed angle - an angle that has a vertex on a circle and sides that contain chords of the circle. 1/2 the measure of the intercepted arc.

inscribed polygon - an inscribed polygon is a polygon in which all vertices lie on a circle.

circumscribed - a circle is circumscribed about a polygon if the circle contains all the vertices of the polygon. circumscribed is in relation to the shape on the "outside".

circumscribed circle - a circle that passes through every vertex of a triangle or regular polygon.

10.2 - Finding Arc & Angle Measures

the measure of a minor arc is the measure of its central angle.

the measure of a major arc is the difference between 360 degrees and the measure of the related minor arc.

arc addition - the measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs. adjacent arcs share a common endpoint.

arc length - the distance between the endpoints along an arc measured in linear units. fraction of the circumference.

equation to find arc length is:

l = (x/360)(2)(pi)(r)

10.3 - Apply Properties of Chords

in the same circle, or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent.

if one chord is a perpendicular bisector of another chord, then the first chord is a diameter.

if a diameter of a circle is perpendicular to a chord, then it bisects the chord and its arc.

in the same circle, or in congruent circles, two chords are congruent if and only if they are equidistant from the center.

10.4 - Use Inscribed Angles & Polygons

an inscribed angle = 1/2 the measure of its intercepted arc.

if two inscribed angles of a circle intercept the same arc, then the angles are congruent.

if a right triangle is inscribed in a circle, then the hypotenuse is the diameter of the circle. this is because the right angle of the triangle is a inscribed angle, whose intercepted arc would be 180 degrees.

a quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary.

10.5 - Tangents

if a line is tangent to a circle then it is perpendicular to the radius drawn to the point of tangency.

if two segments from the same exterior points are tangent to a circle, then they are congruent.

in a plane, a line is tangent to a circle if and only if it is perpendicular to a radius drawn to the point of tangency.

if two segments from the same exterior point are tangent to a circle, then they are congruent.

10.6 - Apply Other Angle Relationships in Circles

angles on the circle theorem - if a tangent and a chord intersect at a point on the circle, then the measure of each angle formed is one-half the measure of its intercepted arc.

to find the angle that is directly tangent -- 1/2(measure of arc it intercepts)

angles inside the circle theorem - if two chords intersect inside a circle, then the measure of each angle is one-half the sum of the measures of the arcs intercepted by the angle and its vertical angle.

to find the angle -- 1/2 (measure of arc it intercepts) + 1/2 (measure of the arc the vertical angle intercepts)

angles outside the circle theorem - if a tangent and a secant, two tangents, or two secants intersect outside a circle, then the measure of the angle formed is 1/2 the difference of the measures of the intercepted arcs.

for tangent & secant, two tangents, and two secants -- 1/2(m. larger arc - m. smaller arc)

these are called power theorems!

10.7 - Find Segment Lengths in Circles

segments of the chord - when two chords intersect in the interior of a circle, each chord is divided into two segments called segments of the chord.

secant segment - a secant segment is a segment that contains a chord of a circle and has exactly one endpoint outside the circle.

external segment - an external segment is the part of a secant segment that is outside the circle.

segments of the chords theorem - if two chords intersect in the interior of a circle, then the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chord.

(chord 1, pt.1)(chord 1, pt.2) = (chord 2, pt.1)(chord 2, pt.2)

segments of secants theorem - if two secant segments share the same endpoint outside a circle, then the product of the lengths of one secant segment and its external segment equals the product of the lengths of the other secant segment and its external segment.

(entire secant segment 1)(external segment 1) = (entire secant segment 2)(external segment 2)

segments of secants and tangents theorem - if a secant segment and a tangent segment share an endpoint outside a circle, then the product of the lengths of the secant segment and its external segment equals the square of the length of the tangent segment.

(entire secant segment)(external segment) = tangent segment squared

10.8 - Equations of Circles

standard equation of a circle:

(x-h)^2 + (y-k)^2 = r^2

center: (h,k) -- radius: r

- or + is inverted because we are squaring it... - moves things to the right, + moves things to the left.

for example: (x-2)^2 + (y+3)^2 = 9

center: (2,-3) -- radius: 3

use distance formula to find the distance between points, can be used to find radius.

distance formula: d = rt. (x2 - x1)^2 + (y2-y1)^2

use midpoint formula to find center of circle.

midpoint formula = ((x1+x2)/2 , (y1+y2)/2)

might need to complete the square to get an equation into the standard equation of a circle.

completing the square:

1) group x's and y's together.

2) get constant on other side of the equation

3) find the numbers that complete the square -- (b/2)^2

4) repeat for y (these numbers will always be positive)

5) add both numbers to other side to keep equality

6) simplify equations (b/2)

7) it's in standard form!