How can we help?

42 terms

Circle

The set of all point in a plane that are equidistant from a given point called the center of the circle. (Page- 651)

Center of a Circle

The exact middle of a circle. (Page- 651)

Radius

A segment whose endpoints are the center and any point on the circle. (Page- 651)

Chord

A segment whose endpoints are on a circle. (Page- 651)

Diameter

A chord that contains the center of the circle. (Page- 651)

Secant

A line that intersects a circle in 2 points. (Page- 651)

Tangent

A line in the plane of a circle that intersects the circles in exactly 1 point, the point of tangency. (Page- 651)

Theorem 10.1

In a plane, a line is tangent to a circle if and only if the line is perpendicular to a radius of the circle at its endpoint on the circle. (Page- 653)

Theorem 10.2

Tangent segments from a common external point are congruent. (Page- 654)

Central Angle

An angle whose vertex is the center of the circle. (Page- 659)

Minor Arc

Part of a circle that measures less then 180°. (Page- 659)

Major Arc

Part of a circle that measures between 180° and 360°. (Page- 659)

Semicircle

An arc with endpoints that are the endpoints of a diameter. (Its 180°). (Page- 659)

Measure of a Minor Arc

The measure of its central angle. (Page- 659)

Measure of a Major Arc

The difference between 360° and the measure of the related minor arc.

Arc Addition Postulate

The measure of an arc formed by 2 adjacent arcs is the sum of the measures of the 2 arcs. (Page- 660)

Congruent Circles

2 circles are congruent if they have the same radius. (Page- 660)

Congruent Arcs

2 arcs are congruent if they have the same measure and they are arcs of the same circle or of congruent circles. (Page- 660)

Theorem 10.3

In the same circle, or in congruent circles, 2 minor arcs are congruent if and only it their corresponding chords are congruent. (Page- 664)

Theorem 10.4

If one chord is a perpendicular bisector of another chord, the first chord is a diameter. (Page- 665)

Theorem 10.5

If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc. (Page- 665)

Theorem 10.6

In then same circle, or in congruent circles, 2 chords are congruent if and only if they are equidistant from the center. (Page- 666)

Inscribed Angle

An angle whose vertex is on a circle and whose sides contain chords of the circle. (Page- 672)

Intercepted Arc

The arc that lies in the interior of an inscribed angle and has endpoints of the angle. (Page- 672)

Measure of an Inscribed Angle Theorem

The measure of an inscribed angle is ½ the measure of its intercepted arc. (Page- 672)

Theorem 10.8

If 2 inscribed angles of a circle intercept the same arc, then the angles are congruent. (Page- 673)

Inscribed Polygon

All the vertices of the polygon lie on the circle. (Page- 674)

Circumscribed circle

The circle that contains the vertices of the inscribed polygon. (Page- 674)

Theorem 10.9

If a right triangle is inscribed in a circle, then the hypotenuse is a diameter of the circle. Conversely, if one side of an inscribed triangle is a diameter of the circle, then the triangle is a right triangle and the angle opposite the diameter is the right angle. (Page- 674)

Theorem 10.10

A quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary. (Page- 675)

Theorem 10.11

If a tangent and a chord intersect at a point of a circle, then the measure of each angle formed is ½ the measure it its intercepted arc. (Page- 680)

Intersecting Lines and Circles

If 2 lines intersect a circle, there are 3 places where the lines can intersect:

1. On the circle

2. Inside the circle

3. Outside the circle

1. On the circle

2. Inside the circle

3. Outside the circle

Angles Inside the Circle Theorem

If 2 chords intersect inside a circle, then the measure of each angle is ½ the sum of the measures of the arcs intercepted by the angle and its vertical angle. (Page- 681)

Angles Outside Theorem

If a tangent and a secant, 2 tangents, or 2 secants intersect outside of a circle, then the measure of the angle formed is ½ the difference of the measures of the intercepted arcs. (Page- 681)

Segments of the Chord

When 2 chords intersect in the interior of a circle, each chord is divided into 2 segments that are called segments of the chord. (Page- 689)

Segments of Chords Theorem

If 2 chords intersect in the interior of a circle, then the product of the lengths of the segments of 1 chord is equal to the product of the lengths of the segments of the other chord. (Page- 689)

Secant Segment

A segment that contains a chord of a circle, and has exactly one endpoint outside the circle. (Page- 690)

External Segment

The part of the secant segment that is outside the circle. (Page- 690)

Segments of Secants Theorem

If 2 segments share the same endpoint outside a circle, then the product of the lengths of 1 secant segment and its external segment equals the product of the lengths of the other secant segment and its external segment. (Page- 690)

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 segments equals the square of the length of the tangent segment. (Page- 691)

Locus

The set of all points in a plane that satisfy a given condition or a set of given conditions. (Page- 697)

Standard Equation of a Circle

The standard equation of a circle with the center (h,k) and the radius r is- (x-h)^2 + (y-k)^2 = r^2 (Page- 699)