Geomety Unit 4b - Notes Tiangle Relationships This unit is boken into two pats, 4a & 4b. test should be given following each pat. quidistant fom two points the same distance fom one point as fom anothe. istance fom a point to a line the length of the pependicula segment fom the point to the line. X = 4.66 cm X = 3.82 cm X = 3.09 cm X = 2.56 cm X = 2.35 cm X It s nice to demonstate how the distance fom the teache to the back wall of the classoom can diffe depending on the angle at which you appoach it. Students can visualize this! s you can see in the diagam above, the shotest distance goes pependiculaly fom X to. Syllabus Objective 4.10 - The student will constuct special segments of a tiangle. ltitude of a Tiangle the pependicula segment fom a vetex of a tiangle to the opposite side o to the line that contains the opposite side. It is helpful to point out: Two of the altitudes of a ight tiangle ae the legs of the ight tiangle. In ode to constuct two of the altitudes in an obtuse tiangle it will equied the extension of the 2 sides of the tiangle that ceate the obtuse angle. onstucting two is sufficient to ceate all thee. The constuction of this segment will equie the pefomance of the pependicula to a line though a point NOT on a line constuction. (See Unit 1 notes) ngle isecto Theoem: If a point is on the bisecto of an angle, then it is equidistant fom the two sides of the angle. onvese of the ngle isecto Theoem: If a point is in the inteio of an angle and is equidistant fom the sides of the angle, then it lies on the bisecto of the angle. ngle isecto of a Tiangle a ay, line o a segment that divides an angle of a tiangle of a tiangle into two adjacent angles that ae conguent. It is helpful to point out: ngle bisectos always intesect inside the tiangle. onstucting two is sufficient to ceate all thee. Unit 4b Tiangle Relationships Page 1 of 14
The constuction of this segment would equie the pefomance of the bisect an angle constuction. (See Unit 1 notes) Median of a Tiangle a segment whose endpoints ae a vetex of the tiangle and the midpoint of the opposite side. It is helpful to point out: Medians always intesect inside the tiangle. onstucting two is sufficient to ceate all thee. The constuction of this segment will equie the pefomance of the bisect a line segment constuction. (See Unit 1 notes) Once the midpoint is detemined the segment connecting the vetex and midpoint can be dawn. Pependicula isecto of a Tiangle a line, ay, o a segment that is pependicula to a side of a tiangle though the midpoint of that side. It is helpful to point out: Pependicula bisectos ae not equied to pass though a vetex! (Though they sometimes do.) The constuction of this segment will equie the pefomance of the bisect a line segment constuction. (See Unit 1 notes) Pependicula isecto Theoem: If a point is on a pependicula bisecto of a segment, than it is equidistant fom the endpoints of the segment. onvese of the Pependicula isecto Theoem: If a point is equidistant fom the endpoints of the segment, than it is on the pependicula bisecto of the segment. Midsegment of a Tiangle - a segment that connects the midpoints of two sides of a tiangle. It is helpful to point out: Midsegments will NVR pass though a vetex! The constuction of this segment will equie the pefomance of the bisect a line segment constuction. (See Unit 1 notes) The midpoint of two sides would be detemined in this manne and then connected to fom the midsegment. Midsegment Theoem: The segment connecting the midpoints of two sides of a tiangle is paallel to the thid side and is half as long. Syllabus Objective 4.12 - The student will exploe the points of concuency and thei special elationships. Unit 4b Tiangle Relationships Page 2 of 14
oncuency of Pependicula isectos of a Tiangle Theoem: The pependicula bisectos of a tiangle intesect at a point that is equidistant fom the vetices of the tiangle. The pependicula bisectos meet at a point of concuency called the cicumcente. x) cute Tiangle Right Tiangle Obtuse Tiangle Location of point: Inside On Outside icumcente icumcente icumcente The distance fom the cicumcente to a vetex is the adius of the cicle that cicumscibes the tiangle. The pefix cicum means aound. onstucting a cicle with the cicumcente as the cente will yield a cicle that is cicumscibed aound the tiangle. cute Tiangle Right Tiangle Obtuse Tiangle icumcente icumcente icumcente xample: pie eating contest is being set up between the sophomoes, junios and senios. Student teams must un fom the stating position to one of the thee tables, eat a pie, and un back to the stating position. The fist team finished wins. fai placement of the stating positions would be at the cicumcente of the tiangle fomed the tables. Unit 4b Tiangle Relationships Page 3 of 14
oncuency of the ngle isectos of a Tiangle Theoem The angle bisectos of a tiangle intesect a point that is equidistant fom the sides of the tiangle. The angle bisectos meet at a point of concuency called the incente. x) cute Tiangle Right Tiangle Obtuse Tiangle Location of point: Inside Inside Inside Incente Incente Incente The distance fom the incente to each side of the tiangle is the adius of the cicle that is inscibed by the tiangle. The pefix in means inside. Notice the incente is always located inside the tiangle. The cicle is inscibed in the tiangle. cute Tiangle Right Tiangle Obtuse Tiangle Incente Incente Incente xample: bicycle patolman must patol thee paths though a pak. These paths fom a tiangle. Whee should he est so that, in case of emegency, he could each any of the thee paths in the shotest peiod of time? The incente. Unit 4b Tiangle Relationships Page 4 of 14
oncuency of the Medians of a Tiangle Theoem: The medians of a tiangle intesect at a point that is two-thids of the distance fom each vetex to the midpoint of the opposite side. The medians meet at a point of concuency called the centoid. Think of the centoid as the cente and the cente is always located inside. Notice, the centoid is also always located inside the tiangle. x) cute Tiangle Right Tiangle Obtuse Tiangle Location of point: Inside Inside Inside entoid entoid entoid **The centoid is 2 3 the distance fom a vetex to the midpoint of the opposite side. xamples: If = 12, then the distance fom to the centoid is 8 and fom the centoid to is 4. If = 6, then the distance fom to the centoid is 4 and fom the centoid to is 2. If = 18, then the distance fom to the centoid is 12 and fom the centoid to is 6. ** The centoid is also the cente of gavity o balancing point of the tiangle. xample: n atist wishes to ceate a metal sculptue showing a tiangle balanced on the tip of an upended cube. Which point should the atist locate on the tiangle? The centoid. Unit 4b Tiangle Relationships Page 5 of 14
oncuency of the ltitudes of a Tiangle Theoem: The lines containing the altitudes of a tiangle ae concuent. The altitudes meet at a point of concuency called the othocente. x) cute Tiangle Right Tiangle Obtuse Tiangle Location of point: Inside On Outside Othocente Othocente Othocente The othocente can be used to find the thid altitude when the othe two ae known. Syllabus Objective 4.11 - The student will apply special segment popeties to solve poblems. xamples: a) 14 and ae midpoints of sides and. is a midsegment of length of leg. Δ. It is 1 2 the 28 b) midsegment of ΔXYZ has a length of ( 3x 1). The side paallel to that midsegment has a length of ( 8x 10). ind the value of x. 2(3x 1) = 8x 10 {x = 4} c) Use the following coodinate plane to find the length and slope of the midsegment M. How does the length and slope of the midsegment M compae to the side of the tiangle? Unit 4b Tiangle Relationships Page 6 of 14
M 4 2-5 5-2 Outline fo pocedue: Step 1: Identify the coodinates of,,, and. Step 2: Use the distance fomula to detemine the length of and. Step 3: Use the slope fomula to detemine the slope fo and. Step 4: nalyze esults and pocess to answe the question. -4 Solution: Step 1: (5, 2); (1, -2); (-2, -1); (0, 1) Step 2: istance omula = ( x x ) + ( y y ) 2 2 2 1 2 1 = 4 2 and =2 2 x x 2 1 Step 3: Slope omula = y y 2 1 has a slope of 1 and has a slope of 1. Step 4: is ½ the length of, so is a midsegment. and both have the slope of 1, since they have equal slopes they ae paallel to each othe. Syllabus Objective 4.9 - The student will solve poblems applying the popeties of tiangle inequalities. ompaing Measuements of a Tiangle Theoem: If one side of a tiangle is longe than anothe side, then the angle opposite the longe side is lage than the angle opposite the shote side. Theoem: If one angle of a tiangle is lage than anothe angle, then the side opposite the lage angle is longe than the side opposite the smalle angle. Unit 4b Tiangle Relationships Page 7 of 14
xample: Identify the longest and shotest sides of a tiangle. Notice they ae opposite the lagest and smallest angle of the tiangle, espectively. 4 7 Since side is lage than side, the angle opposite will be lage than the angle opposite. Theefoe, angle is lage than angle. 100 50 30 m < m < and < < lso, if you know the measue of two angles, then the side opposite the lage angle will be longe than the side opposite the smalle angle. If you know all thee angle measuements, then you will know how the side lengths compae to each othe as well. xteio ngle Inequality Theoem: The measue of an exteio angle of a tiangle is geate than the measue of eithe of the two nonadjacent inteio angles. {xt. Ineq. Th.} m 1 = m + m 1 If 1 equals the sum of two angles, then it follows that each of the two angles ae smalle than 1. Theefoe, m 1> m and m 1> m. Tiangle Inequality Theoem: The sum of the lengths of any two sides of a tiangle is geate than the length of the thid side. { Δ Ineq. Th.} ctivity: ut 4 pieces of pape, sting, ibbon, etc. Make one piece 3 inches, one piece 4 inches, one piece 6 inches, and one piece 8 inches. eate a tiangle with the thee sides, 3, 4, and 6. Tace the tiangle and label the side lengths. Now, next to it, ceate anothe tiangle with the thee sides, 3, 4 and 8. Tace the esult. Notice what has happened. Notice that the second gouping will not fom X a tiangle. The two shot sides must a sum 3 4 3 4 lage than the thid side, i.e., the fist gouping 3 + 4 = 7, which is lage than the 6 Z length of the thid side, 6. The second Y 8 gouping will at most ovelap fo 7 units, but they will neve ceate a tiangle. Unit 4b Tiangle Relationships Page 8 of 14
xamples: Given thee side lengths, detemine if a tiangle can be fomed. 2 2 4 no 3 4 5 yes 3 3 7 no 5 5 9 yes 7 7 7 yes xample: Given two sides of a tiangle, state the ange of lengths fo the thid side. Two sides: 7 and 12. all the thid side x. ccoding to the tiangle inequality: a) 7 + 12 > x b) x + 7 > 12 c) x + 12 > 7 The thid statement is automatically tue. We need only addess the fist two. 1) 7 + 12 > x says 19 > xox < 19 and 2) x + 7 > 12 says x > 5. So 5 < x < 19. Notice the elationship between the bounday values and the oiginal side lengths. The bounday values ae the sum and diffeence of the oiginal side lengths. This allows students to find a patten fo solving this type of question. **When the oat omes In** The Poblem : Something unexpected happens when you pull something towads you that you can t quite see, this is especially tue fo boats. Imagine you ae on a pie, and you ae pulling in a boat that is floating on the wate some way away. The ope comes up ove the edge of the pie, and lies along the pie as you pull it in. s you pull in 10 metes of ope, the boat moves in too, but does it move exactly 10 metes, moe than 10 metes o less than 10 metes? Unit 4b Tiangle Relationships Page 9 of 14
Notes: You can solve this by making up a height fo the jetty and using the Pythagoean Theoem, but it can be much simple than that... x Solution: The ope tavels fom the bough () of the boat up to and along the pie. s the ope is pulled a distance x, the boat moves fom to. Now the ope is tavelling fom (the new position of the boat) up to and along the pie. Thee is no change in the length of the ope, so = x. o any tiangle, the sum of any two sides is geate than the thid, so in Δ, + >. Theefoe by substitution, + ( x ) >. Hence, x > 0 o > x. So the boat comes in futhe than you pull the ope. Hinge Theoem: If two sides of one tiangle ae conguent to two sides of anothe tiangle, and the included angle of the fist is lage than the included angle of the second, then the thid side of the fist is longe than the thid side of the second. onvese of the Hinge Theoem: If two sides of one tiangle ae conguent to two sides of anothe tiangle, and the thid side of the fist is longe than the thid side of the second, then the included angle of the fist is lage than the included angle of the second. ncouage students to identify the two sets of conguent sides and the hinges. s the hinges open wide the distance acoss the opening inceases. xample: ompae the measues of WX and XY. Since m WZX < m YZX, WX < XY. W X Y 51 55 xample: ind the ange of values containing x. ecause 15 is geate than 11, 75 is geate than (2x + 9). So 75 > 2x + 9 66 > 2x 33 > x. Theefoe x < 33. 15 Z 11 75 (2x + 9) Unit 4b Tiangle Relationships Page 10 of 14
Indiect Poof In an indiect poof, one assumes that the statement to be poved is false. One then uses logical easoning to deduce that a statement contadicts a postulate, theoem, o one of the assumptions. Once a contadiction is obtained, one concludes that the statement assumed must in fact be tue. Remembe a contadiction is a pinciple of logic that says, an assumption cannot be both and the opposite of at the same time. Like a numbe cannot be positive and the opposite of positive (negative) at the same time. Indiect poof can be used to pove algebaic concepts, in numbe theoy, and in geometic poof. xamples: State the assumption necessay to stat an indiect poof of each statement. a) If 6 is a facto of n, then 2 is a facto of n. 2 is not a facto of n. b) 3is an obtuse angle. 3 is not an obtuse angle 3is an acute angle o a ight angle. c) x > 5. x is not geate than 5 x 5. d) Points J, K, and L ae collinea. Points J, K, and L ae not collinea. xample: Wite an indiect poof of the statement tiangle can have only one ight angle. ssume a tiangle can have moe than one ight angle. Since by definition a tiangle has thee non-zeo angles, if two of them ae ight angles then those two will total 180. The sum of the 3 inteio angles of a tiangle is 180. If two angles aleady total 180, then the thid would be 0. This contadicts the definition of a tiangle. Theefoe, a tiangle NNOT have moe than one ight angle. OR, a tiangle can have only one ight angle. Unit 4b Tiangle Relationships Page 11 of 14
This unit is designed to follow the Nevada State Standads fo Geomety, S syllabus and benchmak calenda. It loosely coelates to hapte 5 of Mcougal Littell Geomety 2004, sections 5.1 5.6. The following questions wee taken fom the 1 st semeste common assessment pactice and opeational exams fo 2008-2009 and would apply to this unit. Multiple hoice # Pactice xam (08-09) Opeational xam (08-09) 29. Thee towns fom a tiangle on the map below. Thee towns fom a tiangle on the map below. Geometia Spingfield 9 miles 10 miles uclid ule 7 miles Which statement does NOT epesent possible distances between uclid and Geometia?. etween 2 and 7 miles apat.. etween 7 and 9 miles apat.. etween 9 and 16 miles apat.. etween 49 and 81 miles apat. 31. In Δ, is a ight angle and m = 40. Which list shows the sides in ode fom longest to shotest?.,,.,,.,,.,, 32. tiangle has two sides that have lengths of 7 cm and 17 cm. Which could epesent the length of the thid side of the tiangle?. 24 cm. 18 cm. 10 cm. 7 cm ntepise 5 miles Richmond Which statement epesents the possible distance fom ntepise to Spingfield?. etween 1 and 5 miles apat.. etween 5 and 15 miles apat.. etween 15 and 30 miles apat.. etween 30 and 50 miles apat. In Δ, is a ight angle and m = 50. Which list shows the sides in ode fom shotest to longest?.,,.,,.,,.,, tiangle has two sides that have lengths of 4 cm and 14 cm. Which could epesent the length of the thid side of the tiangle?. 3 cm. 10 cm. 17 cm. 18 cm Unit 4b Tiangle Relationships Page 12 of 14
33. The tiangle below contains thee midsegments. The tiangle below contains thee midsegments. 16 x 6 7 11 14 z 9 z x y What ae the values of x, y, and z?. x = 9, y = 22, z = 7. x = 9, y = 11, z = 14. x = 9, y = 22, z = 14. x = 18, y = 11, z = 7 34. In Δ, SR is a midsegment. y What ae the values of x, y, and z?. x = 8, y = 12, z = 7. x = 8, y = 12, z = 14. x = 13, y = 9, z = 10. x = 12, y = 14, z = 8 In Δ I, GH is a midsegment. S Q G J 5 3 12 R I 4 H What is the length of Q?. 34. 26. 17. 13 What is the length of?. 6. 8. 10. 12 Unit 4b Tiangle Relationships Page 13 of 14
35. The tiangle below shows a point of concuency. Lines l, m, and n, ae pependicula bisectos of the tiangle s sides. m The tiangle below shows a point of concuency. The inteio segments ae angle bisectos. l What is the name of the point of concuency in the tiangle?. centoid. incente. othocente. cicumcente 44. In the figue below, ΔKLM Δ. L n What is the name of the point of concuency in the tiangle?. centoid. cicumcente. incente. othocente In the figue below, Δ Δ XYZ. X 8 cm 4 cm 80 K 47 10 cm M 53 45 5 cm Y Which statement must be tue?. = 8cm. = 6cm. m = 53. m = 80 48. Given that Δ GH is an isosceles ight tiangle, what is the measue of an acute angle of the tiangle?. 45. 60. 90. 120 Which statement must be tue?. m X = 45. m Z = 45. YZ = 3cm. XY = 3cm Given that Δ GH is an equilateal tiangle, what is m G?. 30. 45. 60. 90 Z Unit 4b Tiangle Relationships Page 14 of 14