Of A Triangle - Mr Meyers Math
Page 1 of 75.3Medians and Altitudesof a TriangleWhat you should learnGOAL 1 Use properties ofmedians of a triangle.GOAL 2 Use properties ofaltitudes of a triangle.Why you should learn itREFE To solve real-lifeproblems, such as locatingpoints in a triangle usedto measure a person’sheart fitness as inExs. 30–33.AL LIGOAL 1USING MEDIANS OF A TRIANGLEIn Lesson 5.2, you studied two special types ofsegments of a triangle: perpendicular bisectors ofthe sides and angle bisectors. In this lesson, youwill study two other special types of segments ofa triangle: medians and altitudes.AmedianA median of a triangle is a segment whoseendpoints are a vertex of the triangle and themidpoint of the opposite side. For instance, in ABC shown at the right, D is the midpoint ofÆÆside BC. So, AD is a median of the triangle.BDCThe three medians of a triangle are concurrent. The point of concurrency is calledthe centroid of the triangle. The centroid, labeled P in the diagrams below, isalways inside the triangle.Pacute trianglePPright triangleobtuse triangleThe medians of a triangle have a special concurrency property, as described inTheorem 5.7. Exercises 13–16 ask you to use paper folding to demonstrate therelationships in this theorem. A proof appears on pages 836–837.THEOREMTHEOREM 5.7Concurrency of Medians of a TriangleThe medians of a triangle intersect at a point that is two thirds of thedistance from each vertex to the midpoint of the opposite side.If P is the centroid of ABC, then222AP AD, BP BF, and CP CE.333BDPCEFAThe centroid of a triangle can be used as its balancing point, as shown on thenext page.5.3 Medians and Altitudes of a Triangle279
Page 2 of 7FOCUS ONAPPLICATIONS1990REFELAL IcentroidA triangular model of uniform thickness anddensity will balance at the centroid of thetriangle. For instance, in the diagram shownat the right, the triangular model will balanceif the tip of a pencil is placed at its centroid.1890 1790CENTER OFPOPULATIONSuppose the location ofeach person counted in acensus is identified by aweight placed on a flat,weightless map of the UnitedStates. The map wouldbalance at a point that is thecenter of the population.This center has been movingwestward over time.EXAMPLE 1Using the Centroid of a TriangleP is the centroid of QRS shown below and PT 5. Find RT and RP.SOLUTION23Because P is the centroid, RP RT.R13Then PT RT º RP RT.13Substituting 5 for PT, 5 RT, so RT 15.PqTS22Then RP RT (15) 10.33 So, RP 10 and RT 15.EXAMPLE 2Finding the Centroid of a TriangleFind the coordinates of the centroid of JKL.yJ (7, 10)SOLUTIONNYou know that the centroid is two thirdsof the distance from each vertex to themidpoint of the opposite side.ÆChoose the median KN. Find theMÆcoordinates of N, the midpoint of JL .The coordinates of N are6 1010 16 , (5, 8). 3 2 7 , 2 2 2PL(3, 6)K (5, 2)11xFind the distance from vertex K to midpoint N. The distance from K(5, 2) toN(5, 8) is 8 º 2, or 6 units.2Determine the coordinates of the centroid, which is 6, or 4 units up from3Ævertex K along the median KN. INTSTUDENT HELPNEER THOMEWORK HELPVisit our Web sitewww.mcdougallittell.comfor extra examples.280The coordinates of centroid P are (5, 2 4), or (5, 6).Exercises 21–23 ask you to use the Distance Formula to confirm that the distancefrom vertex J to the centroid P in Example 2 is two thirds of the distance from Jto M, the midpoint of the opposite side.Chapter 5 Properties of Triangles
Page 3 of 7GOAL 2USING ALTITUDES OF A TRIANGLEAn altitude of a triangle is the perpendicular segment from a vertex to theopposite side or to the line that contains the opposite side. An altitude can lieinside, on, or outside the triangle.Every triangle has three altitudes. The lines containing the altitudes areconcurrent and intersect at a point called the orthocenter of the triangle.EXAMPLE 3LogicalReasoningDrawing Altitudes and OrthocentersWhere is the orthocenter located in each type of triangle?a. Acute triangleb. Right trianglec. Obtuse triangleSOLUTIONDraw an example of each type of triangle and locate its orthocenter.KBJEAFYWDGPZMLqXCRa. ABC is an acute triangle. The three altitudes intersect at G, a point insidethe triangle.ÆÆb. KLM is a right triangle. The two legs, LM and KM, are also altitudes. Theyintersect at the triangle’s right angle. This implies that the orthocenter is onthe triangle at M, the vertex of the right angle of the triangle.c. YPR is an obtuse triangle. The three lines that contain the altitudes intersectat W, a point that is outside the triangle.THEOREMTHEOREM 5.8Concurrency of Altitudes of a TriangleThe lines containing the altitudesof a triangle are concurrent.Æ ÆHFÆIf AE , BF , and CD are the altitudes of ABC, then the lines AE, BF, and CDintersect at some point H.BAEDCExercises 24–26 ask you to use construction to verify Theorem 5.8. A proofappears on page 838.5.3 Medians and Altitudes of a Triangle281
Page 4 of 7GUIDED PRACTICE Concept Check Vocabulary Check? intersect.1. The centroid of a triangle is the point where the three 2. In Example 3 on page 281, explain why the two legs of the right triangle inpart (b) are also altitudes of the triangle.Skill Check Use the diagram shown and the given information to decide in each caseÆwhether EG is a perpendicular bisector, an angle bisector, a median, or analtitude of DEF.ÆÆ3. DG FGÆEÆ4. EG fi DF5. DEG FEGÆÆÆÆ6. EG fi DF and DG FG7. DGE FGEDGFPRACTICE AND APPLICATIONSSTUDENT HELPExtra Practiceto help you masterskills is on p. 811.USING MEDIANS OF A TRIANGLE In Exercises 8–12, use the figure below andthe given information.ÆÆEP is the centroid of DEF, EH fi DF,DH 9, DG 7.5, EP 8, and DE FE.8Æ8. Find the length of FH.GÆ9. Find the length of EH.Æ10. Find the length of PH.PJ7.5D9HF11. Find the perimeter of DEF.12.EP2LOGICAL REASONING In the diagram of DEF above, .EH3PHPHFind and .EHEPPAPER FOLDING Cut out a large acute, right, or obtuse triangle. Label thevertices. Follow the steps in Exercises 13–16 to verify Theorem 5.7.13. Fold the sides to locate the midpoint of each side.ALabel the midpoints.14. Fold to form the median from each vertex to theSTUDENT HELPmidpoint of the opposite side.HOMEWORK HELP15. Did your medians meet at about the sameExample 1: Exs. 8–11,13–16Example 2: Exs. 17–23Example 3: Exs. 24–26point? If so, label this centroid point.28216. Verify that the distance from the centroid to avertex is two thirds of the distance from thatvertex to the midpoint of the opposite side.Chapter 5 Properties of TrianglesLCMNB
Page 5 of 7xy USING ALGEBRA Use the graph shown.17. Find the coordinates of Q, theyÆmidpoint of MN.P (5, 6)Æ18. Find the length of the median PQ.R219. Find the coordinates of thecentroid. Label this point as T.N (11, 2)œM ( 1, 2)20. Find the coordinates of R, thex10Æmidpoint of MP. Show that theNTNR23quotient is .xy USING ALGEBRA Refer back to Example 2 on page 280.Æ21. Find the coordinates of M, the midpoint of KL.ÆÆ22. Use the Distance Formula to find the lengths of JP and JM.223. Verify that JP JM.3STUDENT HELPLook BackTo construct an altitude,use the construction of aperpendicular to a linethrough a point not on theline, as shown on p. 130.CONSTRUCTION Draw and label a large scalene triangle of the giventype and construct the altitudes. Verify Theorem 5.8 by showing that thelines containing the altitudes are concurrent, and label the orthocenter.24. an acute ABC25. a right EFG with26. an obtuse KLMright angle at GTECHNOLOGY Use geometry software to draw a triangle. Label thevertices as A, B, and C.27. Construct the altitudes of ABC by drawing perpendicular lines throughÆ ÆÆeach side to the opposite vertex. Label them AD, BE, and CF.ÆÆÆÆ28. Find and label G and H, the intersections of AD and BE and of BE and CF.29. Prove that the altitudes are concurrent by showing that GH 0.FOCUS ONCAREERSELECTROCARDIOGRAPH In Exercises 30–33, use the followinginformation about electrocardiographs.The equilateral triangle BCD is used to plot electrocardiograph readings.Consider a person who has a left shoulder reading (S) of º1, a right shoulderreading (R) of 2, and a left leg reading (L) of 3.Right shoulder02 4 230. On a large copy of BCD, plot thereading to form the vertices of SRL.(This triangle is an Einthoven’sTriangle, named for the inventor of theelectrocardiograph.)REFELAL ICARDIOLOGYTECHNICIANINTTechnicians use equipmentlike electrocardiographs totest, monitor, and evaluateheart function.NEER TCAREER LINKwww.mcdougallittell.com31. Construct the circumcenter M of SRL.32. Construct the centroid P of SRL.ÆDraw line r through P parallel to BC.33. Estimate the measure of the acute angleÆB4 4 4 2 20LeftshoulderC0224Leftleg4Dbetween line r and MP. Cardiologistscall this the angle of a person’s heart.5.3 Medians and Altitudes of a Triangle283
Page 6 of 7TestPreparation34. MULTI-STEP PROBLEM Recall the formula for the area of a triangle,1A bh, where b is the length of the base and h is the height. The height of2a triangle is the length of an altitude.a. Make a sketch of ABC. Find CD, the height ofÆthe triangle (the length of the altitude to side AB).Cb. Use CD and AB to find the area of ABC.E15Æc. Draw BE, the altitude to the line containingÆside AC.D12Ad. Use the results of part (b) to find8Æthe length of BE.e.Writing Write a formula for the length of an altitude in terms of the baseand the area of the triangle. Explain. ChallengeSPECIAL TRIANGLES Use the diagram at the right.35. GIVEN ABC is isosceles.ÆAÆBD is a median to base AC.ÆBPROVE BD is also an altitude.D36. Are the medians to the legs of an isoscelesCtriangle also altitudes? Explain your reasoning.37. Are the medians of an equilateral triangle also altitudes? Are they containedin the angle bisectors? Are they contained in the perpendicular bisectors?EXTRA CHALLENGEwww.mcdougallittell.com38.LOGICAL REASONING In a proof, if you are given a median of anequilateral triangle, what else can you conclude about the segment?MIXED REVIEWxy USING ALGEBRA Write an equation of the line that passes throughpoint P and is parallel to the line with the given equation. (Review 3.6 for 5.4)39. P(1, 7), y ºx 340. P(º3, º8), y º2x º 3142. P(4, º2), y º x º 1241. P(4, º9), y 3x 5DEVELOPING PROOF In Exercises 43 and 44, state the third congruencethat must be given to prove that DEF GHJ using the indicatedpostulate or theorem. (Review 4.4)EÆÆ43. GIVEN D G, DF GJHAAS Congruence TheoremÆÆ44. GIVEN E H, EF HJASA Congruence PostulateDFG45. USING THE DISTANCE FORMULA Place a right triangle with legsof length 9 units and 13 units in a coordinate plane and use theDistance Formula to find the length of the hypotenuse. (Review 4.7)284Chapter 5 Properties of TrianglesJB
Page 7 of 7QUIZ 1Self-Test for Lessons 5.1– 5.3Use the diagram shown and the giveninformation. (Lesson 5.1)Æy 24 LJK3yÆHJ is the perpendicular bisector of KL .Æ HJ bisects KHL.4x 93x 251. Find the value of x.H2. Find the value of y.TIn the diagram shown, the perpendicularbisectors of RST meet at V. (Lesson 5.2)6Æ83. Find the length of VT .VÆ4. What is the length of VS ? Explain.5.RBUILDING A MOBILE Suppose youSAwant to attach the items in a mobile sothat they hang horizontally. You wouldwant to find the balancing point ofeach item. For the triangular metal plateshown, describe where the balancingpoint would be located. (Lesson 5.3)ECFGDBÆ ÆÆINTAD , BE , and CF are medians. CF 12 in.NEER TOptimizationTHENNOWAPPLICATION LINKwww.mcdougallittell.comTHROUGHOUT HISTORY, people have faced problems involving minimizing resourcesor maximizing output, a process called optimization. The use of mathematics insolving these types of problems has increased greatly since World War II, whenmathematicians found the optimal shape for naval convoys to avoid enemy fire.MTODAY, with the help of computers, optimization techniques are used inmany industries, including manufacturing, economics, and architecture.1. Your house is located at point H in the diagram. You need to do errandsPHat the post office (P), the market (M), and the library (L). In what ordershould you do your errands to minimize the distance traveled?L2. Look back at Exercise 34 on page 270. Explain why the goalie’s positionon the angle bisector optimizes the chances of blocking a scoring shot.WWII naval convoyThomas Hales proves Kepler’scannonball conjecture.19421611Johannes Kepler proposesthe optimal way to stackcannonballs.1972This Olympic stadiumroof uses a minimumof materials.19975.3 Medians and Altitudes of a Triangle285
So, ADÆis a median of the triangle. The three medians of a triangle are concurrent. The point of concurrency is called the The centroid, labeled Pin the diagrams below, is always inside the triangle. acute triangle right triangle obtuse triangle The medians of a triangle have a special concurrency property, as described in Theorem 5.7.File Size: 695KBPage Count: 7
The triangle has three acute angles that are not all equal. It is an acute triangle. 62/87,21 One angle of the triangle measures 90, so it is a right angle. Since the triangle has a right angle, it is a right triangle. 62/87,21 Since all the angles are congruent, it is a equiangular triangle. 62/87,21 The triangle
acute triangle equiangular triangle right triangle obtuse triangle equilateral triangle isosceles triangle scalene triangle Vocabulary. Holt McDougal Geometry 4-2 Classifying Triangles Recall that a triangle ( ) is a polygon with three sides. Triangles can be classified in
Glencoe/McGraw-Hill 184 Glencoe Geometry Classify Triangles by Sides You can classify a triangle by the measures of its sides. Equal numbers of hash marks indicate congruent sides. If all three sides of a triangle are congruent, then the triangle is an equilateral triangle. If at least two sides of a triangle are congruent, then the triangle is an isosceles triangle.
Name _ 1 Geometry 1 Chapter 4 – Triangle Congruence Terms, Postulates and Theorems 4.1 Scalene triangle - A triangle with all three sides having different lengths. Equilateral triangle - All sides of a triangle are congruent. Isosceles triangle - A tri
Apr 25, 2016 · B. When the triangle was reflected, the base of the resulting triangle lies on the same line as the base of the original triangle. C. When the triangle was reflected, the corresponding sides and angle measures of the resulting triangle are the same as the original triangle. D. When the triangle
The triangle has one obtuse The triangle has all acute angle and no congruent sides. angles and two congruent sides. So, the triangle is an So, the triangle is an obtuse scalene triangle. acute isosceles triangle. Classify the triangle. 1. 45í 45í Exercises 6 –11 2. You can use side lengths and angle measures to classify triangles. Reading
Use the Triangle Midsegment Theorem to fi nd distances. Using the Midsegment of a Triangle A midsegment of a triangle is a segment that connects the midpoints of two sides of the triangle. Every triangle has three midsegments, which form the midsegment triangle. The mid
The XilMailbox library provides the top-level hooks for sending or receiving an inter-processor interrupt (IPI) message using the Zynq UltraScale MPSoC and Versal ACAP IPI hardware. Chapter 1: Xilinx OS and Libraries Overview
