Section 7: Triangles Part 2

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Sectin 7: Triangles Part 2 Tpic 1: Triangle Similarity Part 1... 153 Tpic 2: Triangle Similarity Part 2... 156 Tpic 3: Triangle Midsegment Therem Part 1... 159 Tpic 4: Triangle Midsegment Therem Part 2.

1 Sectin 7: Triangles Part 2 Tpic 1: Triangle Similarity Part Tpic 2: Triangle Similarity Part Tpic 3: Triangle Midsegment Therem Part Tpic 4: Triangle Midsegment Therem Part Tpic 5: Triangle Inequalities Tpic 6: Mre Triangle Prfs Tpic 7: Inscribed and Circumscribed Circles f Triangles Tpic 8: Medians in a Triangle Visit MathNatin.cm r search "Math Natin" in yur phne r tablet's app stre t watch the vides that g alng with this wrkbk! 151

2 The fllwing Mathematics Flrida Standards will be cvered in this sectin: G-C Cnstruct the inscribed and circumscribed circles f a triangle, and prve prperties f angles fr a quadrilateral inscribed in a circle. G-CO Prve therems abut triangles; use therems abut triangles t slve prblems. G-GPE Use crdinates t prve simple gemetric therems algebraically. G-SRT Given tw figures, use the definitin f similarity in terms f similarity transfrmatins t decide if they are similar; explain using similarity transfrmatins the meaning f similarity fr triangles as the equality f all crrespnding pairs f angles and the prprtinality f all crrespnding pairs f sides. G-SRT Use the prperties f similarity transfrmatins t establish the AA criterin fr tw triangles t be similar. G-SRT Prve therems abut triangles. G-SRT Use cngruence and similarity criteria fr triangles t slve prblems and t prve relatinships in gemetric figures. 152

3 Sectin 7: Triangles Part 2 Sectin 7 Tpic 1 Triangle Similarity Part 1 What is the difference between cngruent triangles and similar triangles? Circle the best answer chice t cmplete the sentence belw. Justify yur answer. Similar triangles are always smetimes never cngruent triangles. What infrmatin d we need in rder t determine if tw triangles are similar? Cnsider the diagram f PPPPPP and YYYYYY belw, where PPPPPP ~ YYYYYY. Ø is the type f transfrmatin that results in similar figures. P Y Ø Ø Similarity preserves cngruence f crrespnding. Similarity maintains the prprtinality f crrespnding. O H M U List the crrespnding sides and angles f the triangles abve. Circle the best answer chice t cmplete the sentence belw. Justify yur answer. Cngruent triangles are always smetimes never similar triangles. 153

4 Cnsider the diagrams f MMMMMM and BBBBBB belw. NN YY Cnsider the diagrams belw. D AA MM OO BB C A E 10 R L Determine mm MM. Prve that CCCCCC ~ DDDDDD. Determine mm BB. Cmpare the tw angles. If yu knw that tw pairs f crrespnding angles are cngruent, what must be true f the third pair f angles?!"#$%&'!$! )*+,-./,0+%&%!20*304+ Angle-Angle Similarity (AA~) Criterin If tw angles f ne triangle are cngruent t tw angles f anther triangle, then the tw triangles are similar.!"#$%&'!$! )*+,-./,0+%&%!20*304+ Side-Side-Side Similarity (SSS~) Criterin If the lengths f the crrespnding sides f tw triangles are prprtinal, then the triangles are similar. 154

5 Cnsider the diagram belw. T 6 3 B S 5 A 10 L Suppse that yu have TTTTTT and SSSSSS, and PQ RS = UP VR. Identify the criterin that prves that the tw triangles are similar given each additinal statement. RR EE PQ RS = QU SV Prve that TTTTTT~ LLLLLL. E!"#$%&'!$! )*+,-./,0+%&%!20*304+ Side-Angle-Side Similarity (SAS~) Criterin If the lengths f tw sides are prprtinal and their included angles are cngruent n tw different triangles, then the triangles are similar. 155

6 Sectin 7 Tpic 2 Triangle Similarity Part 2 Cnsider the diagram t the right. RR Let s Practice! 1. An artist is designing a sculpture fr the twn square that will cntain tw triangular slids. The artist wants the triangles in the bases f each slid t be similar. Given: OOOO HHHH Prve: YZ YQ = RS QR HH OO SS EE 70 in 75 in Cmplete the fllwing tw-clumn prf. 24 in. 25 in Statements Reasns 1. OOOO HHHH RRRRRR HH, RRRRRR EE in 22 in AA Similarity Criterin a. Are the triangles similar? Justify yur answer. 4. ZQ = QS YQ QR YQ 6. YQ YQ + ZY YQ = QR QR + RS QR = QR 5. Segment Additin Pstulate 6. b. If the triangles are nt similar, what measurement(s) culd be changed t make them similar? Justify yur answer ZY YQ = 1 + RS QR Subtractin Prperty f Equality 156

7 2. Cnsider the diagram t the right. Given: WWWW SSSS WW II Try It! 3. Cnsider the diagram belw. yy Prve: bc = be cd Rd NN A SS Cmplete the fllwing tw-clumn prf. Statements Reasns GG P Q C R B xx 1. WWWW SSSS 1. Given 2. WWWWWW GGGGGG Alternate Interir Angles Therem AA~ Criterin Prve AAAAAA~ PPPPPP by applying prperties f transfrmatins. Justify yur steps. 5. bc = be cd Rd

8 4. A surveyr is measuring the width f a river fr a future bridge. 64 m 1. Cnsider the figure belw. BEAT THE TEST! T H 9 m 12 m a. What similarity criterin can be used t prve that the triangles are similar? G Which f the fllwing culd be used t prve that HHHHHH and TTTTTT are similar? Select all that apply. N I b. Use the prperties f similar triangles t set up a prprtin and determine the width f the river. NN is the midpint f GGGG TTTT HHHH TT II TTTT IIII HHHH bisects IIII and TTTT TTTTTT is dilated by a scale factr less than 1 centered at pint II. 158

9 2. Mrs. Rbinsn assigned her class a prject t find the height f the flagple. The students culd nt easily measure the height, s they had t use their knwledge f similar triangles t determine the height f the flagple. One student placed a mirrr n the grund 21 feet frm the base f the flagple and backed up until the reflectin f the tp f the ple was centered in the mirrr. Sectin 7 Tpic 3 Triangle Midsegment Therem Part 1 If the midpint f a segment is the halfway pint, then what is the midsegment f a triangle? Cnstruct the midsegment f triangle DDDDDD belw and label the midsegment MMMM. EE FF Part A: If the student is 5.4 feet tall and is standing 7.2 feet frm the mirrr, hw tall is the flagple? DD Part B: Describe anther way t use similarity f triangles t find the height f the flagple.!"#$%&'!$! )*+,-./,0+%&%!20*304+ Triangle Midsegment Therem The midsegment f a triangle jins the midpints f tw sides f a triangle such that its length is half the length f the third side f the triangle and it is parallel t the third side f the triangle. 159

10 Cnsider triangle DDDDDD with midsegment MMMM belw. Let s Practice! EE PP 1. In the diagram belw, AA is lcated at the rigin, GG is lcated at (6, 15), HH is lcated at (6, 9) and II is lcated at (12, 12). Assume that DD, EE, FF, GG, HH, and II are midpints. FF yy MM BB DD GG FF II CC DD HH Describe the relatinship between MMMM and DDDD. AA EE xx Describe the relatinship between DDDD and MMMM. Describe the relatinship between FFFF and PPPP. The area f the triangle MMMMMM is what prtin f the area f the triangle DDDDDD? a. Cmplete the fllwing table. Vertex BB CC DD EE FF Crdinates 160

11 b. Which f the fllwing statements abut the previus diagram are crrect abut the previus diagram? Select all that apply. Triangles AAAAAA and GGGGGG are issceles. Triangle DDDDDD is equilateral. GGGG HHHH GGGG The area f triangle AAAAAA is 16 times larger than the area f triangle GGGGGG. The area f triangle GGGGGG is ne half the area f triangle DDDDDD. FFFF > DDDD If triangle AAAAAA is dilated by a scale factr f s t centered at CC, then AA BB CC CCCCCC. Try It! 2. If the midsegment f an issceles triangle is 5 feet lng and each f the cngruent sides f the triangle are 6 feet lng, what is the measure f the side parallel t the midsegment? 3. The Michigan Triangle is a regin lcated in the middle f Lake Michigan. The triangle is bunded by Bentn Harbr, MI; Ludingtn, MI; and Manitwc, WI. If yu draw their midsegments int the Michigan Triangle, hw des the perimeter f the midsegment triangle cmpare t the perimeter f the Michigan Triangle? 161

12 Sectin 7 Tpic 4 Triangle Midsegment Therem Part 2 Try It! 2. Cnsider the diagram belw. Let s Practice! 1. Cnsider the diagram belw. Given: EEEE, KKKK, and OOOO are midsegments f MMMMMM. EE YY KK YY Prve: The perimeter f EEEEEE = 1 (MMMM + NNNN + YYYY). 2 MM OO NN TT 2xx 14 UU Cmplete the fllwing tw-clumn prf. Statements Reasns OO Find the sum f TTTT and OOOO. xx + 2 LL 1. EEEE, KKKK, and OOOO are midsegments f MMMMMM. 1. Given Triangle Midsegment Therem 3. The perimeter f EEEEEE = (EEEE + KKKK + OOOO) Substitutin 5. The perimeter f EEEEEE = 1 2 (MMMM + NNNN + YYYY)

13 1. Cnsider the figure belw. BEAT THE TEST! II 2. In the fllwing figure, JJJJ is a midsegment f CCCCCC, RRRR is a midsegment f PPPPPP, and OOOO CCCC. PP 45 RR SS PP QQ JJ KK AA 67.5 PP and QQ are the midpints f AAAA and IIII, respectively. Use the prperties f midsegments t find the perimeter f AAAAAA. RR CC Part A: What can yu cnclude abut CC and OO? Justify yur answer. OO Part B: What is the relatinship between the lengths RRRR and CCCC? 163

14 Sectin 7 Tpic 5 Triangle Inequalities Draw a triangle with sides measuring 2 centimeters and 4 centimeters. What is the length f the third side f this triangle? Cnsider the triangle and the infrmatin belw. AAAA + BBBB > AAAA BBBB + AAAA > AAAA AAAA + AAAA > BBBB A C B This is the Triangle Inequality Therem. Draw a triangle with sides 35 millimeters and 18 millimeters. What is the length f the third side f this triangle?!"#$%&'!$! )*+,-./,0+%&%!20*304+ Triangle Inequality Therem The sum f the lengths f any tw sides f a triangle is greater than the length f the third side. What relatinship d yu ntice between the third side and the sum f the first tw sides f the triangle yu drew? Let s Practice! 1. Cnsider the figure belw. A Is it pssible t draw a triangle with a third side that is greater than the sum f the ther tw sides? Why r why nt? C 17 in 12 in B Find the range f pssible sides lengths fr the missing side f the triangle abve. 164

15 Try It! 2. Cnsider the figure belw. Sink!"#$%&'!$! )*+,-./,0+%&%!20*304+ Hinge Therem If tw sides f ne triangle are cngruent t tw sides f anther triangle, and the included angles are nt cngruent, then the lnger third side is ppsite the larger included angle. 7 ft 2 ft Refrigeratr Stvetp Darius is remdeling his kitchen. The kitchen triangle refers t the triangle frmed by the sink, the refrigeratr, and the stvetp. The abve figure represents the remdeling plan. Can the distance frm his sink t stvetp be 9 feet lng? Justify yur answer. Let s Practice! 3. Cnsider the figure belw. 3xx xx Use an inequality t describe a restrictin n the value f xx using the Hinge Therem. 165

16 Try It! 4. Cnsider the figure belw. 5. Cnsider the figure belw. yy 38 4 C 7 D Use an inequality t describe a restrictin n the value f xx using the Hinge Therem. A B E xx In the figure, CCCC = CCCC. DD is a pint n CCCC and EE is a pint n the prlnged line segment CCCC with DDDD = BBBB. Hw wuld yu prve DDDD > BBBB by applying prperties f transfrmatins? Justify yur steps.!"#$%&'!$! )*+,-./,0+%&%!20*304+ Cnverse f the Hinge Therem If tw sides f ne triangle are cngruent t tw sides f anther triangle, and the third side f the first triangle is greater than the third side f the secnd triangle, then the included angle f the first triangle is larger than the included angle f the secnd. 166

BEAT THE TEST! 1. Cnsider the figure belw. Given: ΔAAAAAA with AAAA BBBB Prve: BBBB < AAAA + AAAA A B D C 2. Which f the fllwing culd represent the lengths f the sides f a triangle?

17 BEAT THE TEST! 1. Cnsider the figure belw. Given: ΔAAAAAA with AAAA BBBB Prve: BBBB < AAAA + AAAA A B D C 2. Which f the fllwing culd represent the lengths f the sides f a triangle? Select all that apply. 2 m, 4 m, 7 m 3 m, 8 m, 10 m 9 m, 10 m, 11 m 4 m, 5 m, 9 m 7 m, 13 m, 18 m Use the ptins belw t cmplete the flwchart n the next page t prve the triangle inequality therem. A. The shrtest distance frm a pint qq and a line rr is the line perpendicular t rr and passing thrugh qq. C. Segment Additin D. Given Pstulate E. Definitin f perpendicular lines. B. If aa < bb and cc < dd, then aa + cc < bb + dd. 167

18 Sectin 7 Tpic 6 Mre Triangle Prfs Cnsider the figures belw with AAAAAA DDDDDD. Let s Practice 1. Cnsider the diagram t the right. H B D Given: AAAA KKKK; HHHH bisects AAAAAA Prve: AA KK A W K E Cmplete the fllwing tw-clumn prf. A C F Statements 1. AAAA HHHH 1. Given Reasns List the cngruency statements abut these triangles. Nw, cnsider the fllwing therem. 2. HHHH bisects AAAAAA 2. Given 3. AAAAAA KKKKKK 3. Definitin f angle bisectr!"#$%&'!$! )*+,-./,0+%&%!20*304+ CPCTC Crrespnding parts f cngruent triangles are cngruent. 4. HHHH HHHH 4. Reflexive Prperty 5. AAAAAA KKKKKK AA KK 6. When given a cngruence statement abut tw triangles, hw can yu use CPCTC? We can use CPCTC t justify a cngruence statement f angles r sides when tw triangles are prven cngruent. 168

19 2. Cnsider the figure belw. yy Try It! 3. Cnsider the fllwing diagram. R Given: YZ YQ = RS RQ O S II (2aa, 2cc) Prve: OOOO EEEE H E Cmplete the fllwing tw-clumn prf. B (0, 0) The abve figure shws BBBBBB where LL is the midpint f BBBB and MM is the midpint f IIII. Prve that LLLL BBBB and LLLL = s à BBBB. L M PP (2aa, 0) xx Statements Reasns 1. YZ YQ RQ 1. Given YZ YQ RQ YQ YQ YQ RQ RQ 3. Substitutin 4. = RQ 4. Substitutin YQ 5. ZQ = QS YQ QR 6. RR RR 6. Reflexive Prperty OOOOOO~ HHHHHH RRRRRR HH and RRRRRR EE 9. OOOO EEEE Definitin f Similar Triangles 169

20 4. Cnsider the figure belw. P Q 5. Cnsider the figure belw. K M A B C D Given: mm AA = mm DD, AAAA DDDD, AAAA BBBB CCCC Prve: BBBB CCCC a. Cmplete the fllwing tw-clumn prf. D M F Statements 1. mm AA = mm DD 1. Given 2. AAAA DDDD, AAAA BBBB CCCC 2. Given Reasns Segment Additin Pstulate 4. AAAA + BBBB = DDDD Transitive Prperty 6. CCCCCC BBBBBB CPCTC 8. BBBB CCCC 8. b. Prve BBBB CCCC by applying prperties f transfrmatins. Justify yur steps. Given: KKKK DDDD and KKKK bisects DDDD Prve: KKKK = KKKK Cmplete the fllwing tw-clumn prf. Statements 1. KKKK DDDD, KKKK bisects DDDD 1. Given 2. DDDD FFFF 2. Reasns 3. DDDDDD FFFFFF 3. Definitin f Perpendicular Lines Reflexive Prperty 5. DDDDDD FFFFFF CPCTC 7. KKKK = KKKK

21 BEAT THE TEST! 1. Cnsider the diagram and flw-prf belw. Given: PPPPPP PPPPPP AAAA bisects TTTTTT T S Sectin 7 Tpic 7 Inscribed and Circumscribed Circles f Triangles Cnsider the figure belw. BB Prve: AAAA AAAA A P AA ZZ D PPPPPP PPPPPP PPPPPP PPPPPP L E CC Given 1. AAAA ââââ bisects TTTTTT 2. TTTTTT LLLLLL AAAA ââââ AAAA ââââ What bservatins can yu make abut AAAA? Given Definitin f bisectr AAAA ââââ AAAA ââââ What bservatins can yu make abut BBBB? 3. Use the chices belw t cmplete the flw-prf. A. TTTTTT LLLLLL B. CPCTC C. AAS D. Cngruent Supplements Therem E. Reflexive Prperty What bservatins can yu make abut CCCC? Pint ZZ is called the f the triangle. It is a pint f cncurrency frmed by the intersectin f the three angle bisectrs. 171

22 Let s Practice! 1. Let AAAAAA be an arbitrary triangle with angle bisectrs AAAA, BBBB, and CCCC. BB Try It! 2. Cnsider the figure belw. PP II AA Cmplete the fllwing paragraph that prves that the three angle bisectrs f the internal angles f a triangle are cncurrent. Let s name the bisected angle in AAAAAA, 1 and 2. Then, since BBBB is the angle bisectr f AAAAAA,. Nw, let s cnstruct perpendicular lines segments frm II t AAAA, with the pint f intersectin labeled as MM; frm II t BBBB, with the pint f intersectin labeled as NN; and frm II t AAAA, with the pint f intersectin labeled as PP. Ntice that AAAAAA, BBBBBB, and CCCCCC, because f AAS. Frm these cngruence statements, we can cnclude that IIII IIII, IIII IIII, and IIII IIII, because. Finally, the Prperty f Cngruence helps us prve that IIII IIII IIII. Therefre, the three angle bisectrs f the internal angles f a triangle are. II CC KK a. What bservatins can yu make abut the circle inside the triangle? b. Cmplete the fllwing statement. Pint II is the f bth the triangle and the circle, and it is equidistant t the f the triangle. VV 172

23 Cnsider the figure belw. Let s Practice! nn 3. What is the lcatin f the circumcenter f a right triangle? 4. Where is the circumcenter f an btuse triangle? pp mm What bservatins can yu make abut the circle arund the triangle? What bservatins can yu make abut mm, nn, and pp? Try It! 5. Tina is making part f a wind chime with a flat wden triangular piece f wd. A wire that is anchred at a pint equidistant frm the sides f the triangle suspends the piece f wd. Where is the anchr pint lcated? Cmplete the fllwing statements. Ø Ø A circle that is circumscribed abut a triangle is called a. The center f the circle is the pint f cncurrency f the perpendicular bisectrs f the triangle and is called the. It is als equidistant t the f the triangle. 173

24 BEAT THE TEST! 1. Three straight rads create the bundaries f a park as shwn belw. Cnsider the figure belw. Sectin 7 Tpic 8 Medians in a Triangle AA MM EE CC A develper wants t build sidewalks that are perpendicular bisectrs f each rad and t plant a maple tree where all three sidewalks meet inside the park. Part A: At what pint shuld the develper plant the maple tree? Part B: Suppse the sidewalks are already built. Which f the fllwing therems, pstulates r prperties help us prve that the three sidewalks are cncurrent? A Hinge Therem B Perpendicular Bisectr Therem C Transitive Prperty f Cngruence D Triangle Midsegment Therem Ø Ø Ø Ø Ø A f a triangle is the segment frm a vertex t the midpint f the ppsite side. Every triangle has medians. Draw the ther tw medians in MMMMMM. The intersectin f the three medians is called the, which is the pint f cncurrency fr the medians f a triangle. Each median creates tw triangles. The length frm the vertex t the centrid is the length frm the centrid t the midpint f the side, which yields a rati f. 174

25 Let s Practice! Try It! 1. Gretel is creating a lg fr her cmpany. She started her design as shwn belw. BB 2. Cnsider the figure belw. AA UU FF GG EE II TT RR BB DD CC SS OO Given: DD, EE and FF are all midpints f each f the sides f AAAAAA, and GG is the pint where AAAA, BBBB, and CCCC meet. IIII, UUUU, and OOOO are all medians f BBBBBB, and TT is the centrid. IIII = 2.4 in, BBBB = 1 in, UUUU = 0.7 in Find RRRR, TTTT, OOOO, and UUUU. Prve: AAAA, BBBB, and CCCC are all medians cncurrent at GG and GG is ne-third f the distance frm the ppsite side t the vertex alng the median. 175

26 BEAT THE TEST! 1. Cnsider the triangle belw. yy THIS PAGE WAS INTENTIONALLY LEFT BLANK II AA SS NN DD LL YY xx Prve that the rati f the median frm the pint f cncurrency is 2: 1 using the crdinate gemetry f the medians and triangles similarity. Test Yurself! Practice Tl Great jb! Yu have reached the end f this sectin. Nw it s time t try the Test Yurself! Practice Tl, where yu can practice all the skills and cncepts yu learned in this sectin. Lg in t Math Natin and try ut the Test Yurself! Practice Tl s yu can see hw well yu knw these tpics! 176

Section 6: Triangles Part 1

Section 6: Triangles Part 1 Topic 1: Introduction to Triangles Part 1... 125 Topic 2: Introduction to Triangles Part 2... 127 Topic 3: rea and Perimeter in the Coordinate Plane Part 1... 130 Topic 4: rea