angents Use properties of tangents. Solve problems involving circumscribed polygons. Vocabulary tangent point of tangency are tangents related to track and field events? In July 001, Yipsi oreno of uba won her first major title in the hammer throw at the World thletic hampionships in dmonton, lberta, anada, with a throw of 70.6 meters. he hammer is a metal ball, usually weighing 16 pounds, attached to a steel wire at the end of which is a grip. he ball is spun around by the thrower and then released, with the greatest distance thrown winning the event. S he figure models the hammer throw event. ircle represents the circular area containing the spinning thrower. ay represents the path the hammer takes when released. is tangent to, because the line containing intersects the circle in eactly one point. his point is called the point of tangency. Study ip angent ines ll of the theorems applying to tangent lines also apply to parts of the line that are tangent to the circle. angents and adii odel Use he eometer s Sketchpad to draw a circle with center W. hen draw a segment tangent to W. abel the point of tangency as. hoose another point on the tangent and name it Y. raw WY. hink and iscuss 1. What is W in relation to the circle?. easure WY and W. Write a statement to relate W and WY. 3. ove point Y along the tangent. How does the location of Y affect the statement you wrote in ercise?. easure WY. What conclusion can you make?. ake a conjecture about the shortest distance from the center of the circle to a tangent of the circle. W Y his investigation suggests an indirect proof of heorem 10.9. hapter 10 ircles ndy yons/etty Images
heorem 10.9 If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency. ample: If is a tangent,. ample 1 ind engths is tangent to at point. ind. ecause the radius is perpendicular to the tangent at the point of tangency,. his makes a right angle and a right triangle. Use the ythagorean heorem to find. () () () ythagorean heorem 3 3,, Simplify. ake the square root of each side. ecause is the length of, ignore the negative result. hus,. 3 he converse of heorem 10.9 is also true. heorem 10.10 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. ample: If, is a tangent. Study ip Identifying angents ever assume that a segment is tangent to a circle by appearance unless told otherwise. he figure must either have a right angle symbol or include the measurements that confirm a right angle. ample You will prove this theorem in ercise. Identify angents a. etermine whether is tangent to. irst determine whether is a right triangle by using the converse of the ythagorean heorem. () () () onverse of ythagorean heorem 3 3,, 3 or Simplify. ecause the converse of the ythagorean heorem is true, is a right triangle and is a right angle. hus,, making a tangent to. 3 b. etermine whether is tangent to. Since S, or 8 units. () () () onverse of ythagorean heorem 8,, 8 S 1 6 Simplify. ecause the converse of the ythagorean heorem did not prove true in this case, is not a right triangle. So, is not tangent to. www.geometryonline.com/etra_eamples esson 10- angents 3
ore than one line can be tangent to the same circle. In the figure, and are tangent to. So, () () () and () () (). () () () () () () () () () () Substitution Subtract () from each side. ake the square root of each side. he last statement implies that. his is a proof of heorem 10.10. heorem 10.11 If two segments from the same eterior point are tangent to a circle, then they are congruent. ample: ample 3 You will prove this theorem in ercise 7. Solve a roblem Involving angents ind. ssume that segments that appear tangent to circles are tangent. and are drawn from the same eterior point and are tangent to, so. and are drawn from the same eterior point and are tangent to, so. y the ransitive roperty,. 6 37 8 37 8 3 efinition of congruent segments Substitution dd to each side. Subtract from each side. ivide each side by 8. 37 6 ine angent to a ircle hrough a oint terior to the ircle 1 onstruct a circle. abel the center. raw a point outside. hen draw. onstruct the perpendicular bisector of and label it line. abel the intersection of and as point. 3 onstruct circle with radius. abel the points where the circles intersect as and. raw. is inscribed in a semicircle. So is a right angle, and is a tangent. hapter 10 ircles You will construct a line tangent to a circle through a point on the circle in ercise 1.
Study ip ommon isconceptions Just because the circle is tangent to one or more of the sides of a polygon does not mean that the polygon is circumscribed about the circle, as shown in the second pair of figures. IUSI YS In esson 10-3, you learned that circles can be circumscribed about a polygon. ikewise, polygons can be circumscribed about a circle, or the circle is inscribed in the polygon. otice that the vertices of the polygon do not lie on the circle, but every side of the polygon is tangent to the circle. ample olygons are circumscribed. olygons are not circumscribed. riangles ircumscribed bout a ircle riangle is circumscribed about. ind the perimeter of if. Use heorem 10.10 to determine the equal measures. 19, 6, and. We are given that, so 6 19 or. 19 6 6 19 or 100 he perimeter of is 100 units. efinition of perimeter Substitution 6 19 oncept heck 1. etermine the number of tangents that can be drawn to a circle for each point. plain your reasoning. a. containing a point outside the circle b. containing a point inside the circle c. containing a point on the circle. Write an argument to support or provide a countereample to the statement If two lines are tangent to the same circle, they intersect. 3. raw an eample of a circumscribed polygon and an eample of an inscribed polygon. uided ractice or ercises and, use the figure at the right.. angent is drawn to. ind if 0.. If 13, determine whether is tangent to. 16 6. hombus is circumscribed about and has a perimeter of 3. ind. 3 H pplication 7. IUU pivot-circle irrigation system waters part of a fenced square field. If the spray etends to a distance of 7 feet, what is the total length of the fence around the field? 7 ft esson 10- angents
ractice and pply or ercises 8 11 1 0 17, 18, 3 6 See amples 1, 3 tra ractice See page 77. etermine whether each segment is tangent to the given circle. 8. 9. 16 30 3 10. H 11. K J 1 H 1 K 136 3 10 6 ind. ssume that segments that appear to be tangent are tangent. 1. 13. 1. U 6 1 1 ft 10 8 ft W ft S 7 V 1. cm 16. 17. ( ) ft 8 cm 17 cm 1 ft K m m J H m m 18. 6 in. 19. 0. 3 in. 30 W 1 ft ft Z in. 8 in. S 16 1 V U in. Study ip ook ack o review constructing perpendiculars to a line, see esson 3-6. 1. SUI onstruct a line tangent to a circle through a point on the circle following these steps. onstruct a circle with center. ocate a point on and draw. onstruct a perpendicular to through point.. Write an indirect proof of heorem 10.10 by assuming that is not tangent to. iven:, is a radius of. rove: ine is tangent to. 6 hapter 10 ircles
ind the perimeter of each polygon for the given information. 3.. S 18, radius of 10 S. Y Z. 6. 6(3 ), 1y diameter of 3 10(z ) Z Y y z 7. Write a two-column proof to show that if two segments from the same eterior point are tangent to a circle, then they are congruent. (heorem 10.11) 8. HHY he film in a 3-mm camera unrolls from a cylinder, travels across an opening for eposure, and then is forwarded into another circular chamber as each photograph is taken. he roll of film has a diameter of holding millimeters, and the distance from chamber the center of the roll to the intake of the chamber is 100 millimeters. o the nearest millimeter, how much of the film would be eposed if the camera were opened before the roll had been totally used? 100 mm roll of film stronomy uring the 0th century, there were 78 total solar eclipses, but only 1 of these affected parts of the United States. he net total solar eclipse visible in the U.S. will be in 017. Source: World lmanac SY or ercises 9 and 30, use the following information. solar eclipse occurs when the moon blocks the sun s rays from hitting arth. Some areas of the world will eperience a total eclipse, others a partial eclipse, and some no eclipse at all, as shown in the diagram below. Sun igure not drawn to scale 9. he blue section denotes a total eclipse on that part of arth. Which tangents define the blue area? 30. he pink areas denote the portion of arth that will have a partial eclipse. Which tangents define the northern and southern boundaries of the partial eclipse? 31. II HIKI ind the measure of tangent. plain your reasoning. otal eclipse oon artial eclipse arth 9 www.geometryonline.com/self_check_quiz esson 10- angents 7 ay assey/etty Images
Standardized est ractice tending the esson 3. WII I H nswer the question that was posed at the beginning of the lesson. How are tangents related to track and field events? Include the following in your answer: how the hammer throw models a tangent, and the distance the hammer landed from the athlete if the wire and handle are 1. meters long and the athlete s arm is 0.8 meter long. 33. I I,,, and are tangent to a circle. If 19, 6, and 1, find. 3. ind the mean of all of the numbers from 1 to 1000 that end in. 96 97 98 00 line that is tangent to two circles in the same plane is called a common tangent. ommon internal tangents intersect the segment connecting the centers. ommon eternal tangents do not intersect the segment connecting the centers. k j ines k and j are common internal tangents. m ines and m are common eternal tangents. efer to the diagram of the eclipse on page 7. 3. ame two common internal tangents. 36. ame two common eternal tangents. aintain Your Skills ied eview 37. S ircles are often used in logos for commercial products. he logo at the right shows two inscribed angles and two central angles. If, m 90, m, and m 90, find m and m. (esson 10-) ind each measure. (esson 10-3) 38. 39. 0. 10 K J 10 K 8 K etting eady for the et esson 1. Write a coordinate proof to show that if is the midpoint of in rectangle, then is isosceles. (esson 8-7) UISI SKI Solve each equation. (o review solving equations, see pages 737 and 738.). 3 1 [( 6) 10] 3. 1 [(3 16) 0]. 1 [( 0) 10]. 3 1 [( 10) ] 8 hapter 10 ircles