Geometry Definitions and Theorems. Chapter 9. Definitions and Important Terms & Facts

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1 Geometry Definitions and Theorems Chapter 9 Definitions and Important Terms & Facts A circle is the set of points in a plane at a given distance from a given point in that plane. The given point is the center of the circle and the given distance is the radius. Any segment that joins the center to a point of the circle is called a radius. All radii of a circle are congruent. How to name a circle with its center at A: A chord is a segment whose endpoints lie on a circle. A secant is a line that contains a chord. A diameter is a chord that contains the center of a circle. (Like the word radius, the word diameter can refer to the length of a segment or to a segment.)

2 A tangent is a line in the plane of a circle that intersects the circle in exactly one point, called the point of tangency. A tangent can be a line, a ray, or a segment. A sphere with center O and radius r is the set of all point in space at a distance r from point O. Many of the terms used with spheres are the same as those used with circles. Congruent circles (or congruent spheres) are circles or spheres with have congruent radii. Concentric circles that lie in the same plane and have the same plane and have the same center.

3 Concentric spheres are spheres that have the same center. A polygon is inscribed in a circle and the circle is circumscribed about the polygon when each vertex of the polygon lies on the circle. When each side of a polygon is tangent to a circle, the polygon is said to be circumscribed about the circle and the circle is inscribed in the polygon. A line that is tangent to each of two coplanar circles is called a common tangent. A common internal tangent intersects the segment joining the centers. A common external tangent does not intersect the segment joining the centers.

4 A circle can be tangent to a line, but it can also be tangent to another circle. Tangent circles are coplanar circles that are tangent to the same line at the same points. Circle A and Circle B are externally tangent circles. Circle C and Circle D are internally tangent circles. A central angle of a circle is an angle with its vertex at the center of the circle. The sides of a central angle are radii of the circle. An arc is an unbroken part of a circle. A minor arc is an arc with a measure of less than 180 degrees. A major arc is an arc with a measure of more than 180 degrees.

5 A semicircle measures 180 degrees. The measure of a minor arc is defined to be the measure of its central angle. There are 360 in a circle. Adjacent arcs of a circle are arcs that have exactly one point in common. Congruent arcs are arcs, in the same circle or in congruent circles, that have equal measures. the arc of chord RS :

6 A point Y is called the midpoint of XYZ if XY YZ. Any line, segment, or ray that contains Y bisects XYZ. An inscribed angle is an angle whose vertex is on a circle and whose sides contain chords of the circle. Example of an angle intercepting an arc: Theorems and Corollaries If a line is tangent to a circle, then the line is perpendicular to the radius drawn to the point of tangency.

7 Tangents to a circle from a point are congruent. If a line in the plane of a circle is perpendicular to a radius at its outer endpoint, then the line is tangent to the circle. The measure of the arc formed by two adjacent arcs is the sum of the measures of these two arcs. In the same circle or in congruent circles, two minor arcs are congruent if and only if their central angles are congruent.

8 In the same circle or in congruent circles: (1) Congruent arcs have congruent chords. (2) Congruent chords have congruent arcs. A diameter (or a radius) that is perpendicular to a chord bisects the chord and its arc. In the same circle or in congruent circles: (1) Chords equally distant from the center (or centers) are congruent. (2) Congruent chords are equally distant from the center (or centers). The measure of an inscribed angle is equal to half the measure of its intercepted arc. If two inscribed angles intercept the same arc, then the angles are congruent.

9 An angle inscribed in a semicircle is a right angle. If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary. The measure of an angle formed by a chord and a tangent is equal to half the measure of the intercepted arc. The measure of an angle formed by two chords that intersect inside a circle is equal to half the sum of the measures of the intercepted arcs.

10 The measure of an angle formed by two secants, two tangents, or a secant and a tangent drawn from a point outside a circle is equal to half the difference of the measures of the intercepted arcs.

11 When two chords intersect inside a circle, the product of the segments of one chord equals the product of the segments of the other chord. When two secant segments are drawn to a circle from an external point the product of one secant segment and its external segment equals the product of the other secant segments and its external segment. When a secant segment and a tangent segment are drawn to a circle from an external point, the product of the secant segment and its external segment is equal to the square of the tangent segment.