Geometry, Definitions, Postulates, and Theorems

Geometry, Definitions, Postulates, and Theorems

Chapter 10: Circles

Section 10.1: Tangents to Circle

Circle – The set of all points in a plane that are equidistant from a given point called

the "center." A circle is named by its center point.

Radius – The distance from the center to a point on the circle.

Note: two circles are congruent if they have equal radii.

Diameter – The distance across the circle through the center.

The diameter is twice the radius.

Chord – A segment whose endpoints are points on the circle.

A diameter is the longest chord in a circle.

Secant – A line that intersects a circle in two points.

Tangent – A line in the plane of a circle that intersects the circle in one point.

Tangent Circles – Coplanar circles that intersect at one point.

Concentric circles – Coplanar circles that have a common center (bull’s eye).

Common tangent – A line or segment that is tangent to two coplanar circles.

Common Internal Tangent – A tangent to the circles that crosses the

segment that connects the centers of the circles.

Common External Tangent – A tangent to the circles that does not cross the

segment that connects the centers of the circles.

Interior and Exterior of a circle – The inside and outside of a circle.

Point of tangency – The point at which a tangent line intersects the circle.

Theorem 10.1 – IF a line is tangent to a circle,

THEN it is perpendicular to the radius drawn to the point of tangency.

Theorem 10.2 – In a plane,

IF a line is perpendicular to a radius of a circle at its endpoint on the

circle,

THEN the line is tangent to the circle.

Theorem 10.3 – IF two segments from the same exterior point are tangent to a circle,

THEN the segments are congruent.

Section 10.2: Arcs and Chords

Central Angle – In a plane, an angle whose vertex is at the center of a circle.

Minor Arc – The part cut off by the central angle. (Pac-Man’s mouth)

Major Arc – A major arc measures more than 180°. (Pac-Man’s head)

Semicircle – An arc whose endpoints are the same as those of a diameter

The measure of a semicircle is 180.

Measure of a Minor Arc – The measure of a minor arc equals the measure of the

central angle. (i.e. mÐ = ) The measure of a minor arc is less than 180°.

Measure of a Major Arc – The measure of a major arc equals

360° minus the measure of the associated minor arc.

Arc Addition Postulate – small arc + small arc = big arc

Theorem 10.4– In the same circle, or in congruent circles, two minor arcs are

congruent if and only if their corresponding chords are congruent.

Theorem 10.5 – IF the diameter of a circle is perpendicular to a chord,

THEN the diameter bisects the chord and its arc.

Theorem 10.6 – IF one chord is a perpendicular bisector of another chord,

THEN the first chord is a diameter of teh circle.

Theorem 10.7 – In the same circle, or in congruent circles, two chords are congruent

if and only if they are equidistant from the center.

Section 10.3: Inscribed Angles

Inscribed angle – An angle whose vertex is ON a circle and whose sides contain

chords.

Intercepted arc – The arc that lies in the interior of an inscribed angle and has

endpoints on the angle.

Theorem 10.8 –Measure of an Inscribed Angle

IF an angle is inscribed in a circle (vertex is ON the circle),

THEN its measure is one-half times the measure of its intercepted arc.

(i.e. mÐ = )

Theorem 10.9 – IF two inscribed angles of a circle intercept the same arc,

TEHN the angles are congruent.

Inscribed – A polygon is inscribed in a circle if each vertex of the polygon is on the

circle.

Circumscribed – A polygon circumscribed around a circle if each segment of the

polygon is tangent to the circle.

***Theorem 10.10 – IF a right triangle is inscribed in a circle,

THEN the hypotenuse is a diameter of the circle.

The converse is also true.

Theorem 10.11 – IF a quadrilateral is inscribed in a circle,

THEN the opposite angles are supplementary.

Section 10.4: Other Angle Relationships In Circles

Theorem 10.12 – IF a tangent and a chord intersect at a point ON a circle,

THEN the measure of each angle formed is one half the measure of its intercepted arc. (i.e. mÐ = )

Theorem 10.13 – 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.

(i.e. mÐ = )

Theorem 10.14 – IF a tangent and a secant, two tangents, or two secants intersect the

OUTSIDE a circle,

THEN the measure of the angle formed is one half the difference of

the measures of the intercepted arcs.

(i.e. mÐ = )

Section 10.5: Segment Lengths In Circles

Tangent Segment – A segment that is tangent to a circle at its end point.

Secant segment – A segment with one endpoint on the circle, passes through the

circle, and the other endpoint is outside the circle..

External segment – The portion of a secant segment on the OUTSIDE of the circle..

Theorem 10.15 – IF two chords intersect INSIDE 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. (i.e. )

Theorem 10.16 – IF two segments share the same endpoint OUTSIDE a circle,

THEN the product of the length of one secant segment and the length of its external segment equals the product of the length of the other secant segment and the length of its external segment.

(i.e. : Part Outside · Whole = Part Outside · Whole )

Theorem 10.17 – IF a secant segment and a tangent segment share an endpoint

OUTSIDE a circle,

THEN the product of the length of the secant segment and the

length of its external segment equals the square of the length of the

tangent segment.

(i.e. : Part Outside · Whole = tangent2 )

Section 10.6: Equations of Circles

Standard Equation of a Circle –

Where r is the radius of the circle and (h, k) is the center of the circle.

Section 10.7: Locus

Locus – The set of all points in a plane that satisfy a given condition or a set of given

conditions.