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NOMBRE5FECHAPERÍODOEjercicios preparatoriosRelaciones en TriángulosPaso 1Antes de comenzar el Capítulo 5 Lee cada enunciado. Decide si estás de acuerdo (A) o en desacuerdo (D) con el enunciado. Escribe A o D en la primera columna O si no estás seguro(a) de la respuesta,escribe NS (No estoy seguro(a).PASO 1A, D o NSPASO 2AoDEnunciado1. Cualquier punto ubicado sobre la mediatriz de unsegmento, equidista de los extremos de dicho segmento.2. El circuncentro del triángulo equidista de los puntosmedios de cada lado del triángulo.3. Las alturas de un triángulo se unen en el ortocentro.4. Se pueden dibujar tres alturas para cualquier triángulo.5. La mediana de un triángulo es un segmento que contieneel punto medio de un lado del triángulo.7. El lado más largo en un triángulo está en el lado opuestoal ángulo más pequeño de éste.8. Para escribir una prueba indirecta de que dos rectas sonperpendiculares, comienza por suponer que las dos rectasno son perpendiculares.9. La longitud del lado más largo de un triángulo es siempremayor que la suma de los otros dos lados.10. En dos triángulos, si dos pares de lados son congruentes,entonces la medida de los ángulos inscritos determinaqué triángulo tiene el tercer lado más largo.Paso 2Después de completar el Capítulo 5 Vuelve a leer cada enunciado y completa la última columna con una A o una D. ¿Cambió cualquiera de tus opiniones sobre los enunciados de la primera columna? En una hoja de papel aparte, escribe un ejemplo de por qué estás en desacuerdo con losenunciados que marcaste con una D.Capítulo 54Geometrica de GlencoeCopyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.6. La medida del ángulo exterior de un triángulo es siempremayor que las medidas de cualquiera de sus ángulosinteriores no adyacentes.

NAMEDATE5-1PERIODStudy Guide and InterventionBisectors of TrianglesPerpendicular BisectorPerpendicular BisectorTheoremIf a point is on the perpendicular bisector of a segment, then it is equidistantfrom the endpoints of the segment.Converse of PerpendicularBisector TheoremIf a point is equidistant from the endpoints of a segment, then it is on theperpendicular bisector of the segment.Circumcenter TheoremThe perpendicular bisectors of the sides of a triangle intersect at a point calledthe circumcenter that is equidistant from the vertices of the triangle.Example 1 BD is the perpendicular bisector of AC. Find x.Example 2Find the measure of FM.'C5x - 62.8BCopyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.(,D 3x 8A.AD DC3x 8 5x - 614 2x7 x FK is the perpendicular bisector of GM.FG FM2.8 FMExercisesFind each measure.1. XW2. BF:" 4.2 # '57.5;5198199&&Point P is the circumcenter of EMK. List anysegment(s) congruent to each segment below. 3. MY 4. KP 5. MN 6. ER:1.Chapter 553/,Glencoe GeometryLesson 5-1A perpendicular bisector is a line, segment, or ray that isperpendicular to the given segment and passes through its midpoint. Some theorems dealwith perpendicular bisectors.

NAMEDATE5-1PERIODStudy Guide and Intervention(continued)Bisectors of TrianglesAngle BisectorsAnother special segment, ray, or line is an angle bisector, whichdivides an angle into two congruent angles.Angle BisectorTheoremIf a point is on the bisector of an angle, then it is equidistant from the sidesof the angle.Converse of AngleBisector TheoremIf a point in the interior of an angle if equidistant from the sides of the angle, thenit is on the bisector of the angle.Incenter TheoremThe angle bisectors of a triangle intersect at a point called the incenter that isequidistant from the sides of the triangle. " is the angle bisector of NMP. Find x if m 1 5x 8MRand m 2 8x - 16.ExampleNR1M2PMR! is the angle bisector of NMP, so m 1 m 2.5x 8 8x - 1624 3x8 xCopyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.ExercisesFind each measure.1. ABE2. YBA:"8"&##43 847 ' 4. EWL3. MK87.(7x 5) (3x 21) &52x 1 3x - 8-,)Point U is the incenter of GHY. Find eachmeasure below.5. MU6. UGM7. PHU8. HUChapter 5.1221 51V6(28 #:Glencoe Geometry

NAME5-1DATEPERIODSkills PracticeBisectors of TrianglesFind each measure.1. FG2. KL3'135x - 17(&.4.23x 1-13%Lesson 5-13. TU,4. LYF52x 244- 65x - 30'58 3:5. IU6. MYW(4x - 1) :5(2x 5) Copyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.2x 53619 19 17x*.8"Point P is the circumcenter of ABC. List anysegment(s) congruent to each segment below.3 7. BR41 8. CS# 9. BP(3x 2) 51 (4x - 9) Point A is the incenter of PQR. Find eachmeasure below.20 "11. AU212. QPKChapter 56510. ARU740 ,3Glencoe Geometry

NAMEDATE5-1PERIODPracticeBisectors of TrianglesFind each measure.1. TP2. VU5"907-97x 271 3x 10 63. KN4. NJZ)JI#3x/38 ZNx 10,5. QA6. MFZ.(x 9) (2x - 1) A1E,Point L is the circumcenter of BKT. List anysegment(s) congruent to each segment below.( 7. BN)-# 8. BL/5:Point A is the incenter of LYG. Find eachmeasure below.21 9. ILA"10. JGA-32 (11. SCULPTURE A triangular entranceway has walls with corner angles of 50, 70, and 60.The designer wants to place a tall bronze sculpture on a round pedestal in a centrallocation equidistant from the three walls. How can the designer find where to place thesculpture?Chapter 58Glencoe GeometryCopyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.RQ;'3x 167x

NAME5-1DATEPERIODWord Problem Practice1. WIND CHIME Joanna has a flat woodentriangular piece as part of a wind chime.The piece is suspended by a wireanchored at a point equidistant from thesides of the triangle. Where is theanchor point located?4. NEIGHBORHOOD Amanda is lookingat her neighborhood map. She noticesthat her house along with the homes ofher friends, Brian and Cathy, can be thevertices of a triangle. The map is on acoordinate grid. Amanda’s house is atthe point (1, 3), Brian’s is at (5, -1),and Cathy’s is at (4, 5). Where would thethree friends meet if they each left theirhouses at the same time and walked tothe opposite side of the triangle alongthe path of shortest distance from theirhouse?Copyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.2. PICNICS Marsha and Bill are goingto the park for a picnic. The park istriangular. One side of the park isbordered by a river and the other twosides are bordered by busy streets.Marsha and Bill want to find a spot thatis equally far away from the river andthe streets. At what point in the parkshould they set up their picnic?5. PLAYGROUND A concrete company ispouring concrete into a triangular formas the center of a new playground.a. The foreman measures the triangleand notices that the incenter and thecircumcenter are the same. Whattype of triangle is being created?b. Suppose the foreman changes thetriangular form so that thecircumcenter is outside of thetriangle but the incenter is insidethe triangle. What type of trianglewould be created?3. MOVING Martin has 3 grown children.The figure shows the locations ofMartin’s children on a map that has acoordinate plane on it. Martin would liketo move to a location that is the samedistance from all three of his children.What are the coordinates of the locationon the map that is equidistant from allthree children?5-5Chapter 5Oy5x9Glencoe GeometryLesson 5-1Bisectors of Triangles

NAME5-1DATEPERIODEnrichmentInscribed and Circumscribed CirclesThe three angle bisectors of a triangle intersect in a single point called the incenter. Thispoint is the center of a circle that just touches the three sides of the triangle. Except for thethree points where the circle touches the sides, the circle is inside the triangle. The circle issaid to be inscribed in the triangle.1. With a compass and a straightedge, construct the inscribedcircle for PQR by following the steps below.Step 1 Construct the bisectors of R and Q. Label the pointwhere the bisectors meet, A. Step 2 Construct a perpendicular segment from A to RQ. Usethe letter B to label the point where the perpendicular segment intersects RQ.Step 3 Use a compass to draw the circle with center at A and radius AB .PRQConstruct the inscribed circle in each triangle.2.3.G4. Follow the steps below to construct the circumscribed circlefor FGH. Step 1 Construct the perpendicular bisectors of FG and FH.Use the letter A to label the point where theperpendicular bisectors meet. Step 2 Draw the circle that has center A and radius AF.FHConstruct the circumscribed circle for each triangle.5.Chapter 56.10Glencoe GeometryCopyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.The three perpendicular bisectors of the sides of a triangle also meet in a single point. Thispoint is the center of the circumscribed circle, which passes through each vertex of thetriangle. Except for the three points where the circle touches the triangle, the circle isoutside the triangle.

NAME5-2DATEPERIODStudy Guide and InterventionMedians and Altitudes of TrianglesMedians A median is a line segment that connects a vertex of a triangle to the midpointof the opposite side. The three medians of a triangle intersect at the centroid of thetriangle. The centroid is located two thirds of the distance from a vertex to the midpoint ofthe side opposite the vertex on a median.ExampleIn ABC, U is the centroid and"BU 16. Find UK and BK.2BKBU 3216 BK33,624 BK# 4Lesson 5-2BU UK BK16 UK 24UK 8ExercisesCopyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.In ABC, AU 16, BU 12, and CF 18. Findeach measure.1. UD 2. EU%6Ӏ'3. CU4. AD5. UF6. BE"&In CDE, U is the centroid, UK 12, EM 21,and UD 9. Find each measure.7. CU8. MU9. CK10. JU11. EU12. JDChapter 511 6 12,9.%Glencoe Geometry

NAMEDATE5-2Study Guide and InterventionPERIOD(continued)Medians and Altitudes of TrianglesAltitudesAn altitude of a triangle is a segment from a vertex to the line containing theopposite side meeting at a right angle. Every triangle has three altitudes which meet at apoint called the orthocenter.y# (7, 7)ExampleThe vertices of ABC are A(1, 3),B(7, 7) and C(9, 3). Find the coordinates of theorthocenter of ABC.Find the point where two of the three altitudes intersect."(9, 3) (1, 3)x0Find the equation of the altitude from A to BC. If BC has a slope of 2, then the altitude1has a slope of .2 C to AB. 2If AB has a slope of , then the altitude has a33slope of - .2y - y1 m(x – x1)y - y1 m(x - x1)Point-slope form1y-3 (x – 1)1m , (x1, y1) A(1, 3)2Distributive PropertySimplify.3y - 3 - (x - 9)2273y - 3 - x 22333y - x 223m - , (x1, y1) C(9, 3)2Distributive PropertySimplify.Solve the system of equations and find where the altitudes meet.51y x 2333y - x 22253331 x - x 222533 2x 2222222233Subtract from each side.2 14 2x7 x557511y x (7) 621Subtract x from each side.Divide both sides by -2.2The coordinates of the orthocenter of ABC is (6, 7).ExercisesCOORDINATE GEOMETRY Find the coordinates of the orthocenter of each triangle.1. J(1, 0), H(6, 0), I(3, 6)Chapter 52. S(1, 0), T(4, 7), U(8, 3)12Glencoe GeometryCopyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.211y - 3 x – 2251y x 22Point-slope form

NAME5-2DATEPERIODSkills PracticeMedians and Altitudes of TrianglesIn PQR, NQ 6, RK 3, and PK 4.Find each length.2-1. KM2. KQ4 ,1.3/4. LR5. NK6. PM3In STR, H is the centroid, EH 6,DH 4, and SM 24. Find each length.7. SH4%8. HM&3)Copyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.9. TH10. HR11. TD12. ER5COORDINATE GEOMETRY Find the coordinates of the centroid of each triangle.13. X( 3, 15) Y(1, 5), Z(5, 10)14. S(2, 5), T(6, 5), R(10, 0)COORDINATE GEOMETRY Find the coordinates of the orthocenter of each triangle.15. L(8, 0), M(10, 8), N(14, 0)Chapter 516. D( 9, 9), E( 6, 6), F(0, 6)13Glencoe GeometryLesson 5-23. LK

NAME5-2DATEPERIODPracticeMedians and Altitudes of TrianglesIn ABC, CP 30, EP 18, and BF 39. Find each length.1. PDE2. FPBC1830FPD3. BP4. CD5. PA6. EAAIn MIV, Z is the centroid, MZ 6, YI 18, and NZ 12.Find each measure.7. ZR.8. YZ/9. MR10. ZV11. NV12. IZ:;*3713. I(3, 1), J(6, 3), K(3, 5)14. H(0, 1), U(4, 3), P(2, 5)COORDINATE GEOMETRY Find the coordinates of the orthocenter of each triangle.15. P(-1, 2), Q(5, 2), R(2, 1)16. S(0, 0), T(3, 3), U(3, 6)17. MOBILES Nabuko wants to construct a mobile out of flat triangles so that the surfacesof the triangles hang parallel to the floor when the mobile is suspended. How canNabuko be certain that she hangs the triangles to achieve this effect?Chapter 514Glencoe GeometryCopyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.COORDINATE GEOMETRY Find the coordinates of the centroid of each triangle.

NAME5-2DATEPERIODWord Problem PracticeMedians and Altitudes of Triangles4. MEDIANS Look at the right trianglebelow. What do you notice about theorthocenter and the vertices of thetriangle?Copyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.2. REFLECTIONS Part of the working spacein Paulette’s loft is partitioned in theshape of a nearly equilateral trianglewith mirrors hanging on all threepartitions. From which point couldsomeone see the opposite corner behindhis or her reflection in any of the threemirrors?5. PLAZAS An architect is designing atriangular plaza. For aesthetic purposes,the architect pays special attention to thelocation of the centroid C and thecircumcenter O.a. Give an example of a triangular plazawhere C O. If no such exampleexists, state that this is impossible.3. DISTANCES For what kind of triangle isthere a point where the distance to eachside is half the distance to each vertex?Explain.b. Give an example of a triangular plazawhere C is inside the plaza and O isoutside the plaza. If no such exampleexists, state that this is impossible.c. Give an example of a triangular plazawhere C is outside the plaza and O isinside the plaza. If no such exampleexists, state that this is impossible.Chapter 515Glencoe GeometryLesson 5-21. BALANCING Johanna balanced atriangle flat on her finger tip. What pointof the triangle must Johanna betouching?

NAME5-2DATEPERIODEnrichmentConstructing Centroids and OrthocentersThe three medians of a triangle intersect at a single point called the centroid.You can use a straightedge and compass to find the centroid of a triangle.61. With a straightedge and compass, construct thecentroid for STU by following the steps below.Step 1 Locate the midpoints of sides TU and SU.Label the midpoints A and B respectively.Step 2 Draw the segments SA and TB. Use theletter H to label their point of intersection,which is the centroid of STU.Construct the centroid of each triangle.2.546#")543.The three altitudes of a triangle meet in a single point called the orthocenter of the triangle.Step 1 Extend segments CD and DE past pointD long enough to meet perpendicularsfrom E and C as shown.Step 2 Construct the perpendicular from point Cto the line DE and label the point ofintersection X. Likewise, label the point ofintersection of line CD with the perpendicularfrom E as point Z. In this caseboth X and Z lie outside CDE.Step 3 Label O the point where perpendiculars!# and EZ!# intersect. This is theCXorthocenter of CDE. %&%& 90;Construct the orthocenter of each triangle.5.Chapter 56.16Glencoe GeometryCopyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.4. Follow the steps below to construct the orthocenterof CDE using a straightedge and compass.

NAME5-3DATEPERIODStudy Guide and InterventionInequalities in One TriangleAngle InequalitiesProperties of inequalities, including the Transitive, Addition, andSubtraction Properties of Inequality, can be used with measures of angles and segments.There is also a Comparison Property of Inequality.For any real numbers a and b, either a b, a b, or a b.The Exterior Angle Inequality Theorem can be used to prove this inequality involving anexterior angle.BExterior AngleInequality TheoremThe measure of an exterior angle of a triangleis greater than the measure of either of itscorresponding remote interior angles.1ACDm 1 m A,m 1 m BExampleList all angles of EFG whose measures areless than m 1.G41 2The measure of an exterior angle is greater than the measure ofeither remote interior angle. So m 3 m 1 and m 4 m 1.3FEHUse the Exterior Angle Inequality Theorem to list all ofthe angles that satisfy the stated condition.L31 21. measures are less than m 154JKExercises 1–2MLesson 5-3Copyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.Exercises2. measures are greater than m 33. measures are less than m 1U3 54. measures are greater than m 17X5. measures are less than m 761 42TWExercises 3–8V6. measures are greater than m 2S7. measures are greater than m 58N78. measures are less than m 4Q9. measures are less than m 11R10. measures are greater than m 4Chapter 56217354OExercises 9–10PGlencoe Geometry

NAMEDATE5-3PERIODStudy Guide and Intervention(continued)Inequalities in One TriangleAngle-Side RelationshipsWhen the sides of triangles arenot congruent, there is a relationship between the sides andangles of t

without charge; and be used solely in conjunction with Glencoe Geometry. Any other reproduction, for use or sale, is prohibited without prior written permission of the publisher. Send all inquiries to: Glencoe/McGraw-Hill 8787 Orion Place Columbus, OH 43240 - 4027 ISBN: 978-0-07-890514-8 MHI