Geometry Chapter 10 Study Guide Name

Transcription

1 eometry hapter 10 Study uide Name Terms and Vocabulary: ill in the blank and illustrate. 1. circle is defined as the set of all points in a plane that are equidistant from a fixed point called the center. circle is named by its center 2. segment whose endpoints are the center of a circle and a point on the circle is called a radius. T: ll radii of the same circle are congruent. 3. segment whose endpoints are 2 points on a circle is called a chord. 4. chord that passes through the center of a circle is called a diameter. omparing a diameter to a radius, d 2r 5. The circumference is the distance around a circle. 6. The formula for finding the circumference of a circle is: 2π r or dπ 7. π is defined as the ratio of the circumference of a circle to its diameter. It is approximately equal to central angle angle is an angle whose vertex is the center of a circle and its sides contain radii of the circle. Practice: O pages : 16-19, 26-29, 32-46; The three types of arcs:

2 Type of rc Minor rc Major rc Semicircle xample m on 142 m 142 c 1 m on c m 124 m on c m 180 ow is it named? Name a minor arc with 2 letters: Name a major arc with 3 letters: Name a semicircle with 3 letters: The rc s degree measure is qual to The measure of a minor arc is the same as the measure of its central angle. minor arc ha a measure less than 180 degrees. The measure of a major arc is the 360 minor arc associated with it. major arc has a measure greater than 180 degrees and less than 360. The measure of a semicircle is always 180 degrees. 10. Theorem: In the same circle or congruent circles, two arcs are congruent if and only if their corresponding central angles are congruent. 11. y the way, p if and only if q or p q indicates a statement that is biconditional. It literally means p q and q p. This kind of statement is only true when both p and q have the same truth value. 12. The rc ddition Postulate: The measure of an arc formed by two adjacent arcs is the sum of the measures of the individual arcs. P S Q xample: mpq + mqr mpr R

3 13. rc Length: The length of an arc is a fraction of the circumference of a circle. The fraction is determined by the degree measure of the arc. The following ratio can be used to find. arc measure l ( ircumference) 360 arc measure ( 2π r ) 360 Practice: O: page 533: Theorem: In a circle or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent. Pictured: If then nd If then 15. f the sides of a polygon are chords of a circle, then the polygon is said to be inscribed in the circle and the circle is said to be circumscribed about the polygon. 16. Theorem: In a circle, if a diameter is perpendicular to a chord, the it bisects the chord and it also bisects both of its arcs. Pictured: iven and Then:

4 17. Theorem: In a circle or in congruent circles, two chords are congruent if and only if they are equidistant from the center of the circle. So, In the picture of If If, then and then 18. y the way, the distance from a point to a line is always the length of the segment drawn from the point to the line that is perpendicular to the line. Practice: O page 540: n inscribed angle is an angle that has its vertex on the circle and its sides contain chords. 20. In the picture, M is called the L intercepted arc of LM. xcept for its endpoints and M, the arc lies in the interior of the LM. M 21. Theorem: The measure of an inscribed angle is equal to one half of the measure of its intercepted arc.. Using the picture above: m LM 1 2 ( mm) 22. Theorem: If two inscribed angles of a circle intercept the same arc or congruent arcs, then the angles are congruent. Since and both intercept, they are congruent. Similarly, since is intercepted by and, they are.

5 23. Theorem: If an inscribed angle intercepts or is inscribed in a semicircle, then the angle is a right angle. 24. Theorem: If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary. Pictured: m + m 180 nd m+ m 180 Practice: O Page : 8-10,13-16, tangent line is a line in the plane of a circle that intersects the circle in exactly one point. The point of intersection, (Point in the picture) is called the point of tangency. 26. Theorem: If a line is tangent to a circle then it is perpendicular to the radius drawn to the point of tangency. 27. onverse: If a radius is perpendicular to a line at its endpoint on the circle, then the line is tangent to the circle

6 28. Theorem: If two segments from the same exterior point are tangent to a circle, then the segments are congruent. Practice: O Page : 8,9,12-18, secant is a line that intersects a circle in exactly two points. 30. Theorem: If two secants or chords intersect in the interior of a circle, then the angle formed is half the sum of their intercepted arcs Using the picture to the right: m LM 1 2 ml + mn L M N m MN 1 2 ml + mn 31. Theorem: If a secant and a tangent intersect at the point of tangency, then the measure of each angle formed is half the measure of the intercepted arc. In the picture at the right, N O 1 1 m OPR ( mor) and m QPO mp NO 2 2 Q P R

7 32. Theorem: If two secants, a secant and a tangent, or two tangents intersect in the exterior of a circle, then the measure of the angle formed is equal to one half the positive difference of the two intercepted arcs. Two Secants Secant & Tangent Two Tangents I M L m 1 2 m m mi 1 2 m I mi mm 1 2 ml m Practice: O Page 564: 12-24, 29, Theorem: If two chords intersect in a circle, then the products of the lengths of the segments of each chord are equal rom the picture: MN NP QN NO M Q N O P

8 34. Theorem: If two secant segments are drawn to a circle from an exterior point, then the product of the length of one secant segment and its external secant segment is equal to the product of the other secant segment and its external secant segment. rom the picture: 35. Theorem: If a tangent segment and a secant segment are drawn to a circle from an exterior point, then the square of the length of the tangent segment is equal to the product of the secant segment and its external secant segment. rom the picture: ( ) 2 I I Practice: O Page 572: 8-10, 12,13,15,16,17