* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download The degree measure of an arc is
Survey
Document related concepts
Lie sphere geometry wikipedia , lookup
Riemannian connection on a surface wikipedia , lookup
Line (geometry) wikipedia , lookup
Rational trigonometry wikipedia , lookup
Pythagorean theorem wikipedia , lookup
History of geometry wikipedia , lookup
Perceived visual angle wikipedia , lookup
Approximations of π wikipedia , lookup
Problem of Apollonius wikipedia , lookup
Trigonometric functions wikipedia , lookup
Euclidean geometry wikipedia , lookup
History of trigonometry wikipedia , lookup
Transcript
Geometry Notes C – 1: Circles and Arcs Circles r A circle is the set of points in a plane that are C A circle is usually named by its center: Circle C Facts: All radii of a given circle are Two circles are congruent if Central Angles A central angle of a circle is any angle O Arcs An arc of a circle is B A A minor arc is any arc that is O A major arc is any arc that is C D A semicircle is an arc that is Y A The degree measure of an arc is Note: degree measure is NOT the same as arc length. O Z B Facts: A full circle is A semicircle is Congruent arcs are two arcs of the same circle (or congruent circles) that have B A Congruent central angles intercept congruent arcs and vice versa. C O Arcs that share an endpoint but no other points may be added. D Ex: In circle O, mAB 70 and OB OC . Find A a. mBC b. mAC P B O c. mAPC C Ex: In circle O, mAOB = 70, AB CD and mBC and mAD are in the ratio 3:8. Find mBOC. B C A O D Geometry Notes C – 2: Tangents and Chords Tangents and Chords A A tangent to a circle is a line (in the plane of the circle) that t D intersects the circle in A secant to a circle is a line that intersects the circle in O s D B C l D A chord of a circle is a line segment N Theorem: A tangent and the radius it intersects are A T Given: TAN is tangent to circle O at A; radius OA is drawn Prove: O A tangent segment is a line segment from the point of tangency to another point on the tangent line. Theorem: Two tangent segments drawn to a circle from the same external point Ex: In the diagram at right PA = x x 48 , PB = and 3 x3 A the radius of circle O is 2. P O a. Find the value of x. B b. Find the length of OP . Theorem: If a radius (or diameter) is perpendicular to a chord, then it R B M A O B Theorem: If two chords of a circle are equidistant from the center, then they are , and conversely. C M A O N D Ex: In circle O with radius 12, chord AB is 8 units from O. a. What is the length of the chord? b. What is the degree measure of AB ? Ex: In circle O, PA is a tangent segment and mAB 70 . a. mAOB = A b. mAQ c. mAQO = Q O d. mP = B P Geometry Notes C – 3: Inscribed Angles An inscribed angle of a circle is an angle whose vertex is on the circle and whose sides are contain chords of the circle. Theorem: The measure of an inscribed angle is A B P O Given: Circle O with inscribed angle APB. 1 Prove: mAPO m AB 2 A Case 1: One side of the inscribed angle includes a diameter of the circle. P B O A Case 2: The center of the circle is in the interior of the angle. P O B A B Case 2: The center of the circle is in the exterior of the angle. P O Corollary: If two inscribed angles intercept the same arc, they are congruent. Corollary: Congruent inscribed angles intercept congruent arcs and vice versa. Ex: Find the measure of x in each diagram. a. b. x . c. 70 80 d. x . x . x . 120 Ex: In circle O, AOB is a diameter, mBDC = 30 and mAD : mBD = 5:4. Find 120 C a. mCAB A b. mCBA . E O 30 c. mACD d. mCEB D B Geometry Notes C – 5: Angles Formed by Chords, Secants and Tangents “Interior Angle” D An interior angle of a circle is an angle formed by A E . C B Theorem: The measure of an interior angle in a circle is half the sum of the measures of the arcs intersected by the angle and its vertical angle. D A E C B “Exterior Angle” An exterior angle of a circle is an angle formed by Theorem: The measure of an exterior angle in a circle is half the difference of the measures of the arcs intersected by the angle. A . D P C B Ex: Find the measure of x in each diagram. a. 110 b. . x c. x 80 170 . 40 x . 130 120 d. e. f. 40 40 .70 . 70 . x x x 25 Ex: In the diagram at right, GDF is a tangent and mDA : mBC = 3:4. Find a. mP F D A b. mDEC G E 110 P 20 c. mFDA C B Geometry Notes C – 7: Chords, Secant/Tangent Segments Segments Formed by Intersecting Chords D A Theorem: When two chords intersect inside a circle, E . B C 8 Ex: Solve for x in the diagram. x 6 . Ex: In the diagram (which is not drawn to scale), AE = 12, EB = 6 and CD = 22. Find the lengths of CE and ED if CE > ED. 12 B C . A E D Segments Formed by Intersecting Secants (or a Secant and a Tangent) Vocabulary: PA and PC are called secant segments. PB and PD are called external segments of the secants. A Theorem: When two secants intersect outside a circle, B P . D C Proof: A B P . D C 12 6 Ex: Solve for x in the diagram. . 8 x A Ex: In the diagram (which is not drawn to scale), PA is a tangent segment. If PA = 4 6 and MC is 4 more than PM, find the length of PC . . P E M C Geometry Notes C – 10: Proofs The following facts/theorems may be helpful on tonight’s homework and Thursday’s test. 1. All radii of a circle B 2. A radius (or diameter) and a tangent to a circle A O B 3. The arcs between two parallel chords are C A D 4a. Inscribed angles that intercept congruent arcs b. Inscribed angles that intercept the same arc C C D E B B D F A A 5. An inscribed angle in a semicircle is C B O A