Geometry: A Complete Course

Transcription

1 eometry: omplete ourse with rigonometry) odule - tudent Worket Written by: homas. lark Larry. ollins 4/2010

2 or ercises 20 22, use the diagram below. 20. ssume is a rectangle. a) f is 6, find. b) f is, find. c) f m = 40, find m. d) f m = 6, find m. 21. ssume is a rhombus. a) f m =, find m. b) f m = 86, find m. 22. ssume is a square. a) f = + 6 and = 4 10, find. b) ind m. 2. f quadrilateral is a rectangle, with coordinates 2,), 2,1), 7,1), and 7,), then find the lengths of the diagonals and verify that they are congruent. 24. n quadrilateral, determine the coordinates of point, so that is a square, given that is, 1), is 1,), is, 1), and is,y). 2. uadrilateral is a parallelogram, with = 2 + 4, = 11, and = + 1. how that is a rhombus. 26. iven: is a parallelogram; m = m 4 rove: is a rhombus iven: is a rectangle, is a parallelogram rove: is isosceles 470 nit ther olygons

3 nscribed ngle of a ircle n angle formed by any two chords with a common endpoint. ormally, in our eometry, an angle is an inscribed angle of a circle, if and only if, the verte of the angle is on the circle, and the sides are chords of the circle. angent Line of a ircle rom the Latin word, tangere, meaning, to touch, a tangent line of a circle is a line which intersects the circle in eactly one point, called the point of tangency, or the point of contact. ote: We sometimes refer to a line segment as a tangent segment, if that segment is contained in a tangent line and intersects the circle in such a way that the point of tangency is one of its endpoints. ntercepted rc of a ircle n angle of a cirlce intercepts an arc of a circle, if and only if, each of the following conditions hold: 1) he endpoints of the arc lie on the sides of the angles. 2) ach side of the angle contains one endpoint of the arc. ) ll points on the arc, ecept the endpoints, lie in the interior of the angle. easure of an rc of a ircle ased on its relationship with the central angles of a circle, this is defined as the measure of the central angle which intercepts the arc. easure of a emicircle ecause we can consider a semicircle to be the intercepted arc of a central angle of 180, its rays are radii of the same diameter), we say that the measure of a semicircle is 180. orollary 68a f, in a circle, a diameter is drawn to a tangent line, at the point of tangency, then that diameter is perpendicular to the tangent line, at that point. ample 1: n the figure at the right, is a tangent line to circle at point. f is a secant line intersecting at point, and m is 84, find m. olution: 1 m = m 2 heorem 68) 1 84 = m = 2 1 m = m 42 nit ircles

4 7. n the figure to the right, is tangent to and at points and respectively. lso, = 6, = 8, and = 0. ind,, and. int: se the ythagorean heorem) 8. n the figure above, is tangent to at point and omplete the following statements.. a) is the geometric mean between and. b) is the geometric mean between and. c) f = 6 and = 24, = and =.. n the figure to the right, is a tangent segment to, and is a radius of. f = 17 and = 7, find the length of the radius of. J se the figure to the right for eercises 10 and iven: at center. rove: m = iven: is tangent to and at point K rove: m = 1 /2 m 12. n shown to the right, is a tangent line at point. m = 4 and m = 8. ind m, m, and the measure of each angle of. art ngle and rc elationships 4

5 se the figure to the right, and the given information for eercises 1 through 2. is tangent to at point. is a secant of intersecting the circle at points and. m = 18, m = 2 ote: m > m) 1. ind m 14. ind m 1. ind m 16. ind m 17. ind m 18. ind m 1. ind m 20. and are. 21. ind m 22. ind m 2. ind m 24. ind m 2. ind m 4 nit ircles

6 8 nit ircles 14. ind the value of in. 1. ind the value of on. 16. ind the measure of. 17. ind the measure of. 18. ind the measure of. 1. ind the measure of W. 20. ind the measure of,, and. 21. ind the measure of and. 4 4 W J W 4 J 4 W J = 7 = 4 = 24 = 1 = = 18 mw = 288 W = 14 = 7 m = 8 = 18 = 12 J = 10 m = 140 iven: iven: iven: iven: iven: iven:

7 Lesson 4 ercises: 1. rove heorem 77 - f two secant segments are drawn to a circle from a single point outside the circle, the product of the lengths of one secant segment and its eternal segment, is equal to the product of the lengths of the other secant segment and its eternal segment. ote: his is the same theorem we proved in the lesson. We are using it as an eercise to make sure you understand its proof. se your ourse otes to check.) a) tate the theorem. b) raw and label a diagram to accurately show the conditions of the theorem. c) List the given information. d) Write the statement we wish to prove. e) emonstrate the direct proof using the two column format. 2. rove heorem 78 - f a secant segment and a tangent segment are drawn to a circle, from a single point outside the circle, then the length of that tangent segment is the mean proportional between the length of the secant segment, and the length of its eternal segment. se the outline given in ercise 1. n eercises through 14, find the value of nit ircles

8 11. he perpendicular bisectors of the sides of meet at point. = 11 and = 4. ind. ive a reason for your answer. 12. is given. erpendicular bisectors of the sides,, and are shown. an you conclude that =? f not, eplain, and state a correct conclusion that can be deduced from the diagram. 1. raingle is given. ngle bisectors,, and are shown. an you conclude that =? f not, eplain, and state a correct conclusion that can be deduces from the diagram. 14. he three perpendicular bisectors of the sides of a triangle are concurrent in a point which can be inside the triangle, on the triangle, or outside the triangle. ketch an obtuse triangle, an acute triangle, and a right triangle showing the perpendicular bisectors of the sides in each to verify each relationship. 1. riangle is an obtuse triangle. erpendicular bisectors of the sides meet at point. = 12 and = 1. ind. n eercises 16 through 20, complete the statement using always, sometimes, or never. 16. perpendicular bisector of a side of a triangle passes through the midpoint of a side of the triangle. 17. he angle bisectors of the angle of a triangle intersect at a single point. 18. he angle bisectors of the angle of a triangle meet at a point outside the triangle. 1. he perpendicular bisectors of the sides of a triangle meet at a point which lies outside the triangle he midpoint of the hypotenuse of a right triangle is equidistant from all vertices of the triangle.. art ircles and oncurrency 8

9 y) 2a,0) 6y) 0,2b) 2) 6y) 6y) 2) 4) Lesson 2 ercises: 1. rove heorem 84 - f the opposite angles of a quadrilateral are supplementary, then the quadrilateral is cyclic. onverse of orollary 67b) 4) 2c,0) 2. ind the values of, y, and z.. ind the values of, y, and z. m = 16 m = z z y 0,2b) 12,8) 102 y 4. ind the values of, y, and z.. ind the values of and y. m = z 2a,0) y 11 6y) 2a,0) y 0,2b) 2c,0) 2c,0) 0,0) 6. ind the values of and y. 7. ind the measure of the angles of quadrilateral. 16,0) y y 0,0) 0,0) 12,8) 12,8) 16,0) 16,0) y y y y 11 26y 21y y y y 26y 4 21y W y rove that trapezoid inscribed in a circle, is an isosceles trapezoid. 10. uppose that is a quadrilateral inscribed 14 26y in a circle, and that is a diameter of the circle. y f m is three times m, what are the measure 48 of all four angles? W 24y y 21y W 4 art ircles and oncurrency 40 8