5-. Plan Objectives o identif properties of perpendicular bisectors and angle bisectors o identif properties of medians and altitudes of a triangle amples inding the ircumcenter Real-orld onnection inding engths of edians Identifing edians and ltitudes 5- hat ou ll earn o identif properties of perpendicular bisectors and angle bisectors o identif properties of medians and altitudes of a triangle... nd h o find a location in a backard for the largest possible swimming pool, as in ample oncurrent ines, edians, and ltitudes heck kills ou ll Need GO for Help esson -7 or ercises, draw a large triangle. onstruct each figure.. ee back of book.. an angle bisector. a perpendicular bisector of a side * ). raw GH. onstruct ' GH at the midpoint of GH. * ) * ) * ) * ). raw with a point not on. onstruct '. New Vocabular concurrent point of concurrenc circumcenter of a triangle circumscribed about incenter of a triangle inscribed in median of a triangle centroid altitude of a triangle orthocenter of a triangle ath ackground he theorems in this lesson can be related to eva s heorem, which Giovanni eva published in 678: et sides,, and of be divided at,, and respectivel. hen,, and are concurrent if and onl if?? =. oncurrenc theorems will be applied later to inscribed and circumscribed circles and to the stud of centroids in phsics. ore ath ackground: p. 56 esson Planning and Resources ee p. 56 for a list of the resources that support this lesson. PowerPoint ell Ringer Practice heck kills ou ll Need or intervention, direct students to: onstructing Perpendicular isectors esson -7: ample tra kills, ord Problems, Proof Practice, h. onstructing ngle isectors esson -7: ample 5 tra kills, ord Problems, Proof Practice, h. Properties of isectors. he bisectors of the ' of a k meet at a point inside the k.. he # bis. of the sides of a k intersect at a point that might fall inside, outside, or on the k. 7 hapter 5 Relationships ithin riangles Hands-On ctivit: Paper olding isectors raw and cut out five different triangles: two acute, two right, and one obtuse. tep : Use paper folding to create the angle bisectors of each angle of an acute triangle. hat do ou notice about the angle bisectors? tep : Repeat tep with a right triangle and an obtuse triangle. oes our discover from tep still hold true? olding a Perpendicular isector pecial Needs or ample, have students cop the diagram. Using a compass, have them choose other centers and draw circles as large as possible that lie within the triangle. he will not find a larger circle. olding an ngle isector. ake a conjecture about the bisectors of the angles of a triangle. ee left. tep : Use paper folding to create the perpendicular bisector of each side of an acute triangle. hat do ou notice about the perpendicular bisectors? tep : Repeat tep with a right triangle. hat do ou notice?. ake a conjecture about the perpendicular bisectors of the sides of a triangle. ee left. elow evel Have students use a compass or algebra to confirm that point (, ) is the center of the circle that contains points O, P, and in ample. 7 learning stle: tactile learning stle: tactile
Ke oncepts heorem 5-6 Vocabular ip he prefi circum is atin for around or about. hen three or more lines intersect in one point, the are concurrent. he point at which the intersect is the point of concurrenc. or an triangle, four different sets of lines are concurrent. heorems 5-6 and 5-7 tell ou about two of them. he perpendicular bisectors of the sides of a triangle are concurrent at a point equidistant from the vertices. heorem 5-7 he bisectors of the angles of a triangle are concurrent at a point equidistant from the sides. ou will prove these theorems in the eercises. his figure shows #QR with the perpendicular bisectors of its sides concurrent at. he point of concurrenc of the perpendicular bisectors of a triangle is called the circumcenter of the triangle. Points Q, R, and are equidistant from, the circumcenter. he circle is circumscribed about the triangle. Q Q R R. each Guided Instruction Hands-On ctivit tudents ma construct the angle bisectors and perpendicular bisectors using techniques the learned in esson -7. eaching ip hen discussing heorems 5-6 and 5-7, emphasize that the point of concurrenc is equidistant from vertices for perpendicular bisectors and equidistant from sides for angle bisectors. P onnection to lgebra Remind students that the equation of a horizontal line is = a and the equation of a vertical line is = a. Quick heck P inding the ircumcenter oordinate Geometr ind the center of the circle that ou can circumscribe about #OP. wo perpendicular bisectors of sides of #OP are = and =. hese lines intersect at (, ). his point is the center of the circle. a. ind the center of the circle that ou can circumscribe about the triangle with vertices (0, 0), (-8, 0), and (0, 6). (, ) b. ritical hinking In ample, eplain wh it is not necessar to find the third perpendicular bisector. hm. 5-6: ll of the # bis. of the sides of a k are concurrent. P O (, ) P iversit Remember that some students have little or no eperience with houses that have ards big enough to hold a swimming pool. PowerPoint dditional amples ind the center of the circle that circumscribes. (, ) his figure shows #UV with the bisectors of its angles concurrent at I. he point of concurrenc of the angle bisectors of a triangle is called the incenter of the triangle. I I I I O Points,, and are equidistant from I, the incenter. he circle is inscribed in the triangle. U V esson 5- oncurrent ines, edians, and ltitudes 7 dvanced earners Have students investigate eva s heorem and how it can be used to prove heorem 5-8. learning stle: verbal nglish anguage earners he terms circumscribe and inscribe can be related to their prefies: circum- meaning around and inmeaning within. tudents also need to understand the difference between collinear and concurrent. learning stle: verbal 7
PowerPoint dditional amples it planners want to locate a fountain equidistant from three straight roads that enclose a park. plain how the can find the location. Highwa 0 ariposa oulevard Park ndover Road ocate the fountain at the point of concurrenc of the angle bisectors of the triangle formed b the three roads. Guided Instruction Quick heck a. raw segments connecting the towns. uild the librar at the inters. pt. of the # bisectors of the segments. edians and ltitudes P Real-orld onnection Pools he Jacksons want to install the largest possible circular pool in their triangular backard. here would the largest possible pool be located? ocate the center of the pool at the point of concurrenc of the angle bisectors. his point is equidistant from the sides of the ard. If ou choose an other point as the center of the pool, it will be closer to at least one of the sides of the ard, and the pool will be smaller. a. he towns of damsville, rooksville, and artersville want to build a librar that is damsville? equidistant from the three towns. race the diagram and show where the should build rooksville the librar. ee left. artersville b. hat theorem did ou use to find the location? he # bisectors of the sides of a k are concurrent at a point equidistant from the vertices. median of a triangle is a segment whose endpoints are a verte and the midpoint of the opposite side. edian actile earners tudents can use paper-folding techniques to find altitudes and medians of triangles here and in ercise 5. onnection to Phsical cience Have students read the orling Kindersl (K) ctivit ab on pages 0 0, and do the ctivit involving the centroid as a point of balance. eaching ip he proofs of heorems 5-8 and 5-9 are postponed until students have the tools necessar to complete them. P ath ip Point out that another wa to state heorem 5-8 is that each median is broken into segments that have a ratio of :. his can help students use mental math to find lengths. sk: If =, what does equal? 5 Ke oncepts heorem 5-8 est-aking ip If ou don t remember the meaning of a term, like centroid, the diagram ma give a clue. Quick heck he medians of a triangle are concurrent at a point that is two thirds the distance from each verte to the midpoint of the opposite side. = J = G = H In a triangle, the point of concurrenc of the medians is the centroid. he point is also called the center of gravit of a triangle because it is the point where a triangular shape will balance. (ee K ctivit ab, page 0.) ou will prove heorem 5-8 in hapter 6. P inding engths of edians Gridded Response In # at the left, is the centroid and = 6. ind. ince is a centroid, = and =. = = 6 ubstitute 6 for. = 8 ind. heck that + =. G H J 8. / /... 0 0 0 5 5 5 5 6 6 6 6 7 7 7 7 8 8 8 8 9 9 9 9 7 hapter 5 Relationships ithin riangles 7
or: oncurrent ines ctivit Use: Interactive etbook, 5- Quick heck n altitude of a triangle is the perpendicular segment from a verte to the line containing the opposite side. Unlike angle bisectors and medians, an altitude of a triangle can be a side of a triangle or it ma lie outside the triangle. cute riangle: ltitude is inside. P Right riangle: ltitude is a side. Identifing edians and ltitudes Is a median, an altitude, or neither? plain. is a segment etending from verte to the side opposite. lso, ' VU. is an altitude of #VU. Obtuse riangle: ltitude is outside. Is U a median, an altitude, or neither? plain. edian; U is a segment drawn from verte U to the midpt. of the opp. side. he lines containing the altitudes of a triangle are concurrent at the orthocenter of the triangle. proof of this theorem appears in hapter 6. V U P rror Prevention tudents ma think that and U meet at the centroid or orthocenter of VU. Point out that since is an altitude and U is a median, their point of intersection cannot be categorized. PowerPoint dditional amples is the centroid of OR, and = 6. ind. O R Is K a median, an altitude, neither, or both? K Ke oncepts heorem 5-9 he lines that contain the altitudes of a triangle are concurrent. RI or more eercises, see tra kill, ord Problem, and Proof Practice. Practice and Problem olving GO Practice b ample for Help ample (page 7) oordinate Geometr ind the center of the circle that ou can circumscribe about each triangle.. (, ). O 6 O oordinate Geometr ind the center of the circle that ou can circumscribe about k.. (0, 0). (0, 0) 5. (-, 5) 6. (-, -) 7. (, ) (, 0) (, 0) (-, 5) (-5, -) (, ) (, ) (, -) (-, -) (-, -7) (6, ) (, ) (, ) (, ) (, ) (, ) esson 5- oncurrent ines, edians, and ltitudes 75 (0, 0) both Resources ail Notetaking Guide 5- ail Notetaking Guide 5- dapted Instruction losure Use the diagram above to eplain wh the following must be true: he bisector of the verte angle of an isosceles triangle is both an altitude and a median. he bisector of the verte angle of an isosceles triangle is the perpendicular bisector of the base b heorem -5. ecause the bisector is perpendicular, it is an altitude. ecause it bisects the opposite side, it is a median. 75
. Practice ssignment Guide -0, 7-9,,, 9- ample (page 7) Name the point of concurrenc of the angle bisectors. 8. 9. 5-6, 0,,, 5-8, hallenge -6 est Prep 7- ied Review -5 Homework Quick heck o check students understanding of ke skills and concepts, go over ercises,,, 8, 9. lternative ethod ercise tudents ma trace and cut out the triangle and use paper folding, or carefull construct the perpendicular bisectors on graph paper, to find the point of intersection. ercises 7 If students use graph paper to draw the triangles, the will easil find the horizontal and vertical perpendicular bisectors. 0. it Planning op the diagram of ltgeld Park. how where park officials should place a drinking fountain so that it is equidistant from the tennis court, the plaground, and the volleball court. ind the # bisectors of the sides of the k formed b the tennis court, the plaground, and the volleball court. hat point will be equidistant from the vertices of the k. ample (page 7) ample (page 75) 6. ltitude; is a segment drawn from a verte of a k perp. to the opp. side. ppl our kills In kuv, is the centroid. U 8; 7. If = 9, find and.. If U = 9, find and U. ; U. If V = 9, find V and. V 6; Is a median, an altitude, or neither? plain.. 5. 6. Plaground Volleball ourt Neither; it s not a segment drawn edian; is a midpt. ee left. from a verte. onstructions raw the triangle. hen construct the inscribed circle and the circumscribed circle. 7 8. ee margin. 7. right triangle, # 8. obtuse triangle, #U ltgeld Park ennis ourt V GP Guided Problem olving nrichment Reteaching dapted Practice Practice Name lass ate Practice 5- oncurrent ines, edians, and ltitudes ind the center of the circle that circumscribes kn.... N 6 8 N N. onstruct the angle bisectors for. hen use the point of concurrenc to construct an inscribed circle. Is a perpendicular bisector, an angle bisector, an altitude, a median, or none of these? 5. 6. 7.. ind the circumcenter of the triangle formed b the three pines. In ercises 9, name each figure in k. 9. an angle bisector 0. a median ). a perpendicular bisector. an altitude G. ritical hinking centroid separates a median into two segments. hat is the ratio of the lengths of those segments? : or :. riting Ivars found a ellowed parchment inside an antique book. It read: rom the spot I buried Olaf s treasure, equal sets of paces did I measure; each of three directions in a line, there to plant a seedling Norwa pine. I could not return for failing health; now the hounds of Haiti guard m wealth. Karl fter searching aribbean islands for five ears, Ivars found one with three tall Norwa pines. How might Ivars find where Karl buried Olaf s treasure? G Pearson ducation, Inc. ll rights reserved. 8. 9. 0. or each triangle, give the coordinates of the point of concurrenc of (a) the perpendicular bisectors of the sides and (b) the altitudes.... 5 6 76 hapter 5 Relationships ithin riangles 76
Problem olving Hint Paper-folding an altitude is the same as paper-folding the perpendicular to a line through a point not on the line. 0. It is given that is on line / and line m. the l isect. hm., 5 and 5. the rans. Prop. of 5, 5 5. is on ra n b the onv. of the l is. hm. GO nline Homework Help Visit: PHchool.com eb ode: aue-050 he figures below show how to construct medians and altitudes b paper folding. o find an altitude, fold the triangle so that a side overlaps itself and the fold contains the opposite verte. o find a median, fold one verte to another verte. his locates the midpoint of a side. 5 6. heck students work. 5. ut out a large triangle. Paper-fold ver carefull to construct the three medians of the triangle and demonstrate heorem 5-8. 6. ut out a large acute triangle. Paper-fold ver carefull to construct the three altitudes of the triangle and demonstrate heorem 5-9. 7. ultiple hoice is the centroid of #. If G 5 6 9, what epression represents? 9 6 9 6 hen fold so that the fold contains the midpoint and the opposite verte. 8. Is a perpendicular bisector, an angle bisector, a median, an altitude, or none of these? plain. b. None of these; it is a midsegment. a. b. c. ltitude; is # to a side from a verte. l bisector; it bisects an l. 9. eveloping Proof omplete this proof of GP heorem 5-6 b filling in the blanks. m Given: ines O, m, and n are perpendicular n bisectors of the sides of #. is the intersection of lines / and m. Prove: ine n contains point, and = =. Proof: ince O is the perpendicular bisector of a. 9, =. ince m is the perpendicular bisector of b. 9, = c. 9. hus = =. ince =, is on line n b the onverse of the d. 9 heorem. b. c. d. # bis. Proof 0. Prove heorem 5-7. Given: Ras O, m, and n are bisectors of the angles of #. is the intersection of ras O and m and ', ', '. Prove: Ra n contains point, and = =. n m. hat kind of triangle has its circumcenter on one of its sides? plain. right triangle; check students eplanations. H G rror Prevention! ercise 0 tudents ma not realize that the plaground and courts locate points. iscuss as a class wh these particular points on the plaground and courts might have been chosen. ercise 5 Point out that meets onl half the conditions to be an altitude and onl half the conditions to be a median, which means that it is neither. ercise 7 atch for students who think is G instead of G. sk: Is larger or smaller than G? larger ercises 9, 0 ecause using properties from two segments to prove concurrence is a new and sophisticated idea, discuss these proofs as a class after students complete them. ncourage students to ask questions about the strateg chosen for each proof. onnection to iscrete ath ercise 6 uler (pronounced oiler ) is also responsible for the even ridges of Königsberg problem, the proof of which was fundamental to the development of graph theor. Have students research uler s contributions to mathematics. 7. 8. U esson 5- oncurrent ines, edians, and ltitudes 77 77
. ssess & Reteach PowerPoint esson Quiz. omplete the sentence: o find the centroid of a triangle, ou need to draw at least 9 median(s). two. GH has vertices (, ), G(9, ), and H(9, 0). ind the center of the circle that circumscribes GH. (, ) Use the diagram for ercises 5. P V N. Identif all medians and altitudes drawn in PV. P and are medians; V is an altitude.. If = 5, find and. 0 and 5 5. If =, find P and P. P 8 and P. oordinate Geometr omplete the following steps to locate the centroid. Problem olving Hint a. ind the coordinates of midpoints,, (, 6) ou can prove and N. (, ); (5, ); N(, 0) 6 * ) * ) * ) heorem 5-8 for a b. ee b. ind equations of, N, and. general n with below c. ind the coordinates of P, the intersection coordinates (0, 0), * ) * ) left. of and N. his is the centroid. 0 (b, d), and (c, 0) * ) ( P,) b following the steps d. how that point P is on. ee left. for the particular e. Use the istance ormula to show that n in ercise. point P is of the distance from each N 6 (8, 0) verte to the midpoint of the opposite side. ee margin. hallenge or ercises and, points of concurrenc have been drawn for two triangles. atch the points with the lines and segments listed in I IV. b. : ;.. N 5 I-; II-; III-; IV- : ± ; : ± 7 7 0 d. 7( ) ± 0 7 7 ± 7 7 I-; II-; III-; IV- 5. nswers ma var. ample: et k be I. perpendicular bisectors of sides II. angle bisectors isosc. with base ' III. medians IV. lines containing altitudes and. If bisects l, then it is # to, and therefore the altitude * ) from l. o, contains the circumcenter, incenter, centroid, and orthocenter. est Prep 5. In an isosceles triangle, show that the circumcenter, incenter, centroid, and orthocenter can be four different points but all four must be collinear. ee left. 6. Histor In 765 eonhard uler proved that for an triangle, three of the four points of concurrenc are collinear. he line that contains these three points is known as uler s ine. Use ercises and to determine which point of concurrenc does not necessaril lie on uler s ine. l bisectors lternative ssessment P is a point inside. Have students work in pairs to write a full description of the properties of point P if it is the circumcenter, incenter, centroid, or orthocenter of. est Prep Resources or additional practice with a variet of test item formats: tandardized est Prep, p. 0 est-aking trategies, p. 96 est-aking trategies with ransparencies ultiple hoice hort Response tended Response 78 hapter 5 Relationships ithin riangles Use the figure at the right for ercises 7 9. 7. hat is R if R = 5 cm?. 8 cm. 08 cm. 6 cm. 6 cm R 8. hat is if J = 0 mm? H J. 70 mm G. 05 mm H. 0 mm J. 57.5 mm 9. hat is if = 5 and J = 5 +?. 0.. 0.. 0.6.. 0. Name all tpes of triangles for which the centroid, circumcenter, incenter, and orthocenter are all inside the triangle. lassif the triangles according to the sides as well as the angles. ee margin.. he point of concurrenc of the three altitudes of a triangle lies outside the triangle. here are its circumcenter, incenter, and centroid located in relation to the triangle? raw and label a diagram to support each of our answers. ee back of book. 78
ied ied Review Review GO for Help esson 5-. No; point is not necessaril equidistant from the sides. 9. and 50. and esson - esson - etermine whether point must be on the bisector of l. plain.... 0 5 es; es; point is equidistant bisects the l. from the sides. lassif each kjk b its angles. 5. m&j = 7, m&k = 5, m& = 90 6. m&j = 7, m&k = 98, m& = 5 right obtuse In the figure at the right, is a square. Identif each of the following. 7 5. nswers ma var. * ) * ) * ) * ) 7. a line skew to 8. a line skew to 9. two intersecting planes 50. two parallel segments * ) 5. the intersection of plane and plane heckpoint Quiz essons 5- through 5-5. k O k; H 7. bisects l; ) is equidist. ) from and. 8. ; k Ok b H, so b P. lgebra ind the value of... 6. a. is a midsegment of #. = 5. ind. 0 b. = 6 and = 6. ind the perimeter of #. 8 Use the diagram. hat can ou conclude about each of the following? plain.. & right l; supp. to l 5. # and # 6. and O ; P Use the figure at the right. ) 7. hat can ou conclude about? plain. 8. ind. Justif our response. 7 8. ee left. riting or a given triangle, describe how ou can construct the following. 9. a median 9 0. ee margin. 0. an altitude 0 ee left. Use this heckpoint Quiz to check students understanding of the skills and concepts of essons 5- through 5-. Resources Grab & Go heckpoint Quiz. e. "; P 6 "; Ä 9 N "0 "0; P 60 "0; Ä 9 "58; P "58 Ä 9 0. [] an acute k; or a list that contains all of the following: equiangular >, equilateral >, acute isosceles >, acute scalene > [] a list that does not contain equiangular >, equilateral >, acute isosceles >, or scalene > heckpoint Quiz 9. nswers ma var. ample: isect a side of a k. onnect the opp. verte with the midpt. 0. Use the procedure for constructing a # to a line from a point not on the line. lesson quiz, PHchool.com, eb ode: aua-050 esson 5- oncurrent ines, edians, and ltitudes 79 79