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Name LESSON Date Class Reteach 5-4 The Triangle Midsegment Theorem A m idsegment of a triangle joins the midpoints of two sides of the triangle. Every triangle has three midsegments. _ _ R is the midpoint of _ CD . RS is a midsegment of ᭝C DE. S is the midpoint of CE . _ Use the figure for Exercises 1–4. AB is a midsegment of ᭝R ST . _ 1. What is _ the slope of midsegment AB and the slope of side ST ? Ϫ1; Ϫ1 _ � � � _ _ _ 2. What can you conclude about AB and ST ? Since the slopes are the same, AB ʈ ST. � �� �� �� 3. Find AB and ST . ᎏ ᎏ AB ϭ 2͙ 2 , ST ϭ 4͙ 2 _ _ 4. Compare the lengths of AB and ST . AB ϭ __1 ST or ST ϭ 2AB 2 Use ᭝MNP for Exercises 5–7. _ 5. UV is a midsegment of ᭝MNP. Find the coordinates of U and V. �� U ( Ϫ 1, 3), V (3, 2) _ � _ _ 6. Show that UV ʈ MN . The slope of UV ϭ Ϫ __1 and the slope of 4 _ 1 __ MN ϭ Ϫ . Since the slopes are the _4 _ same, UV ʈ MN. 1 MN 7. Show that U V ϭ __ . ᎏ 2 ᎏ ᎏ � �� ᎏ UV ϭ ͙ 17 and MN ϭ 2͙ 17 . Since ͙ 17 ϭ __1 (2͙ 17 ), UV ϭ __1 MN . 2 2 Copyright © by Holt, Rinehart and W inston. All rights reserved. 30 Holt Geometry Name LESSON Date Class Reteach 5-4 The Triangle Midsegment Theorem continued Theorem Example Triangle Midsegment Theorem A midsegment of a triangle is parallel to a side of the triangle, and its length is half the length of that side. _ Given: PQ is _ a midsegment of ᭝LMN. _ Conclusion: PQ ʈ LN , PQ ϭ __12 LN You can use the Triangle Midsegment Theorem to find various measures in ᭝ABC . 1 AC HJ ϭ __ ᭝ Midsegment Thm. 2 1 (12) HJ ϭ __ Substitute 12 for AC . 2 HJ ϭ 6 Simplify. 1 AB J K ϭ __ 2 1 __ 4 ϭ AB 2 8 ϭ AB _ � _ ᭝ Midsegment Thm. HJ || AC Midsegment Thm. Substitute 4 for J K. mЄBCA ϭ mЄBJH Corr. д Thm. Simplify. mЄBCA ϭ 35° Substitute 35° for mЄBJH. Find each measure. 8. VX ϭ 23 9. HJ ϭ 54 92° 10. mЄVXJ ϭ 11. XJ ϭ 27 Find each measure. 12. ST ϭ 72 13. DE ϭ 22 14. mЄD ES ϭ 48° 15. mЄRCD ϭ 48° Copyright © by Holt, Rinehart and W inston. All rights reserved. 31 Holt Geometry N ame LESSON D ate Class N ame Practice A 5-4 The Triangle Midsegment Theorem U se t he Triangle Midsegment Theorem to name parts of the figure for Exercises 1–5. 5-4 The Triangle Midsegment Theorem _ _ DE _ DE _ AD _ DE _ 1. a mids egment of ABC 2. a s egment parallel to AC � _ 3. a s egment that has the s ame length as BD _ 4. a s egment that has half the length of AC _ � 6. U s e the Midpoint Formula to find the c oordinates of G . ( 7. U s e the Midpoint Formula to find the c oordinates of H . ( _ 0 3 0 0 8. U s e the Slope Formula to find the s lope of D F. _ 9. U s e the Slope Formula to find the s lope of GH . 10. If tw o s egments hav s ame s lope, then the s egments _ e the _ are parallel. Are DF and GH parallel? U se t he Triangle Midsegment Theorem and the figure for Exercises 14–19. Find each measure. 12 15. QR 12 17. m � SU P 55° 19. m � PR Q 55° 125° 27 N ame 9. H ow does the perimeter of the mids egment triangle c ompare to the perimeter of the Bermuda Triangle? It is half the perimeter of the Bermuda Triangle. _ _ _ 10. Given: U S , ST , and TU are mids egments of PQR . Prove: The perimeter of STU _1_(PQ � QR � R P ) . 2 Possible answer: Ho lt Ge o me tr y 28 Copyright © by Holt, Rinehart and Winston. All rights reserved. N ame Practice C 5-4 The Triangle Midsegment Theorem Reasons 1. Given 2. Midsegment Theorem 3. Definition of perimeter D ate Class Ho lt Ge o me tr y Reteach 5-4 The Triangle Midsegment Theorem LESSON LESSON A midsegment of a triangle joins the midpoints of tw o s ides of the triangle. Ev ery triangle has three mids egments . ( 0, 2 ) � ( 0, 0) 2a b 2. Find the c oordinates of the midpoints D , E , and F. Write a two-column proof that the perimeter of a midsegment triangle is half the perimeter of the triangle. 4. Substitution 4. The perimeter of S TU _1_ PQ � _1_ QR 2 2 � _1_ R P. 2 5. The perimeter S TU _1_ (PQ � QR � RP ) 5. Distributive Property 2 of Class Pedro has a hunch about t he area of midsegment triangles. H e is a careful student, so he investigates in a methodical manner. First Pedro draws a right triangle because he knows it will be easy to calculate the area. 1. Find the area of ABC . San J u an 3045 mi 22 D ate Bermuda M iami 1522.5 mi 16. PU � Dis t. ( mi) M iami t o S an J uan 1038 M iami t o B e rmuda 1042 B e rmuda t o S an J uan � 965 Statements _ _ _ 1. US , ST , and TU are midsegments of P QR. 2. ST _1_ PQ , TU _1_ QR , US _1_ RP 2 2 2 3. The perimeter of S TU ST � TU � US. 14. ST 8. Find the perimeter of the mids egment triangle w ithin the Bermuda Triangle. Copyright © by Holt, Rinehart and Winston. All rights reserved. ) 7. U s e the dis tanc es in the c hart to find the perimeter of the Bermuda Triangle. 18. m � SUR , ) yes 6 3 yes 11. U s e the D is tanc e Formula to find D F. 12. U s e the D is tanc e Formula to find G H. 13. Does G H _1_ DF ? 2 2 2 , 18. 2 58° 58° 6. m � D The B ermuda Triangle is a region in t he A tlantic Ocean off the southeast coast of the U nited States. The triangle is bounded by Miami, Florida; San Juan, Puert o R ico; and B ermuda. In t he figure, t he dot t ed lines are midsegments. � 17. 5 58° 4. m � H IF 122° 5. m � H GD � 9.1 3. GE BC 5. a s egment that has tw ic e the length of EC C omplete Exercises _6–13 to show t_ hat midsegment GH is parallel t o DF and that G H _1_DF. 2 Class U se the figure for Exercises 1–6. Find each measure. 9.1 35 1. H I 2. DF D ate Practice B LESSON R S is a mids egment of C DE. R is the midpoint of CD _ . S is the midpoint of C E . D (0, b), E (a, b), F (a, 0) 3. Pedro k now s it w ill be eas y to find the area of EFD if � D EF is a right angle. Write a proof that � D EF � � A . _ _ Possible answer: F is the midpoint of AC, so AF _1_ AC . DE _1_ AC 2 _ _ _ _2 by the Midsegment Theorem, so AF � DE. DE AF by the Midsegment Theorem and E DF _ and � A FD are alternate interior angles, so � E DF _� � � A FD. DF � DF by the Reflexive Property, thus E FD � A DF. By CPCTC, � D EF � � A . _1_ ab 2 4. Find the area of EFD . U se the figure for Exercises 1–4. AB is a midsegment of R ST. _ 1. W hat is _ the s lope of mids egment AB and the s lope of s ide ST ? 1 ; 1 _ _ _ _ 2. What c an y ou c onc lude about AB and ST ? Since the slopes are the same, AB ST. � �� � � � �� � � 5. C ompare the areas of ABC and EFD . 3. Find AB and ST . � A BC has four times the area of E FD. � AB 2 2 , ST 4 2 _1_ ab 2 6. Pedro has already s hown that EFD � AD F. C alc ulate the area of AD F . _ _ ( 2 , 0) _ _ 4. C ompare the lengths of AB and ST . AB _1_ ST or ST 2A B 2 7. Write a c onjec ture about c ongruent triangles and area. Possible answer: Congruent triangles have equal area. Use MN P for Exercises 5–7. _ Pedro already knows some things about the area of the midsegment t riangle of a right t riangle. B ut he thinks he can expand his theorem. B efore he can get to that, however, he has to show another property of triangles and area. 8. Find the area of W XY , W XZ , and YXZ . 16; 6; 10 5 . U V is a mids egment of MN P . Find the c oordinates of U and V . _ The slope of UV _1_ and the slope of 4 MN _1_. Since the slopes are the _4 _ same, UV MN. _ Possible answer: The total of the areas of W XZ and Y XZ is equal to the area of W XY. � �� � � � UV 17 and M N 2 17 . Since 17 _1_ (2 17 ), UV _1_ MN . 2 2 Possible answer: The area of a larger triangle is the sum of areas of the triangles within it. Copyright © by Holt, Rinehart and W inston. All rights reserved. 7. Show that U V _1_MN . � 2 10. Write a c onjec ture about the areas of triangles w ithin a larger triangle. 29 � _ _ 6. Show that U V MN . 9. C ompare the total of the areas of W XZ and YXZ to the area of W XY . Copyright © by Holt, Rinehart and Winston. All rights reserved. �� U (1 , 3), V (3, 2) Ho lt Ge o me tr y Copyright © by Holt, Rinehart and Winston. All rights reserved. 73 30 Ho lt Ge o me tr y Holt Geometry N ame LESSON D ate Class N ame Reteach LESSON 5-4 The Triangle Midsegment Theorem c ontinued Theorem 5-4 Consider Different Cases J K _1_ AB 2 4 _1_ AB 2 8 AB � HJ || AC Mids egment Thm. C orr. Thm. Simplify. m � BC A 35° Subs titute 35° for m � BJ H. 23 case 1: EF FG 7 and E G 8; case 2: EF FG 8 and E G 6 12. ST 72 13. D E 22 Two cases; the midsegment joins the sides with lengths 12 and 18. The midsegment joins the side with lengths 12 and x . 14. m � D ES 48° 15. m � R CD 48° 31 Copyright © by Holt, Rinehart and Winston. All rights reserved. D ate Class Ho lt Ge o me tr y Problem Solving 5-4 The Triangle Midsegment Theorem A midsegment t riangle is formed from the three mids egments of a triangle. 5x � 2 _ _ _ LM ; MN ; NL ( 4, 8) 2. What is the mids egment triangle in Q R S ? 6 (1, �1) 8 9.2 mi _ 3. Whic h mids egment is parallel to s ide QS ? _ MN 4mi 1. 5mi _ 5. LN � U se the diagram for Exercises 7 and 8. 6 cm _ _ QM � MR � 6. D raw the mids egments in ABC . 6. In triangle H JK, m �H 110, m � J 30°, and of _ m � K 40. If R is the midpoint _ J K, and S is the midpoint of H K, w hat is m � J RS ? F 150 G 140 _ _ 4. If s ide R S is 12 c m, how long is LM ? C hoose the best answer. C 240 c m2 D 480 c m2 The Triangle Midsegment Theorem: A mids egment of a triangle is parallel to a s ide of the triangle, and its length is half the length of that s ide. (12, �3) � 4. The diagram at right s how s hors_ ebac k riding trails . Point B is the halfw ay point t D is the _along path AC . Po in _ _halfw ay point along path C E . The paths along BD and AE are parallel. If riders trav el from A to B to D to E , and then bac k to A , how far do they trav el? 1. 7mi L MN 0 5. R ight triangle FGH has mids egments of length 10 c entimeters , 24 c entimeters , and 26 c entimeters . What is the area of FGH ? 1. N ame the mids egments in Q R S . _ Ho lt Ge o me tr y Class LESSON 9.5 Yes; X is the midpoint of LN , and _ Y is the midpoint of ML. D ate A midsegment of a triangle is a s egment that joins the midpoints of tw o s ides of the triangle. 1. The v ertic es _ of J KL are J ( 9 , 2), K (10, 1), 2. In Q R S , QR 2x � 5, R S 3x 1, and_L ( 5, 6). CD is the mids egment and SQ 5x . W hat is the perimeter of _ parallel to J K . W hat is the length of C D ? R ound to the mids egment triangle of Q R S ? the neares t tenth. 3. Is XY a mids egment of L MN if its endpoints are X (8, 2.5) and Y (6.5, 2 )? Ex plain. 32 Copyright © by Holt, Rinehart and Winston. All rights reserved. N ame Reading Strategies 5-4 Identify Relationships LESSON H 110 J 30 _ On the balanc e beam, V is _the midpoint of AB , and W is the midpoint of YB . _ 7. The length of VW is 1 _7 _ feet. W hat is AY ? 8 A _7_ ft C 3 _3 _ ft 8 4 15 ___ B ft D 7 _1 _ ft 16 2 8. The meas ure of � AYW is 50 . W hat is the meas ure of � VW B ? F 45 H 90 G 50 J 130 Copyright © by Holt, Rinehart and Winston. All rights reserved. If the midsegment joins the sides with lengths 12 and 18, then the third side is 18. If the midsegment joins the side with lengths 12 and x, then it is impossible to find the length of the third side. N ame A 60 c m 2 B 120 c m2 4. Find the length of the third s ide of ABC by c ons idering both c as es . 3. How many c as es are there to c ons ider w hen mak ing a c onc lus ion about the third s ide of the triangle? Ex plain. 18 12 � 27 Find each measure. Use A BC for Exercises 4 and 5. A midsegment of t he t riangle is 9. 92° 10. m � VXJ 2. Find the lengths of the triangle’s s ides for eac h of the c as es in Ex erc is e 1. m � BC A m � BJ H 54 � Mids egment Thm. 9. HJ � Subs titute 4 for J K. 8. VX Find each measure. 11. XJ _ _ _ midsegment connects Case 2: The the base EG and one of the congruent sides of E FG. Case 1: The midsegment connects the two congruent sides E F and F G. _ 1. Des c ribe two pos s ible c as es and mak e a drawing of eac h. Given: PQ is a mids egment of L MN . _ _ C onclusion: PQ LN , PQ _1_ LN 2 Triangle EFG is an is os c eles triangle _ w ith EF FG and w ith the perimeter equal to 22 units . A mids egment, QR , o f EFG is equal to 4 units . You c an us e the Triangle Mids egment Theorem to find v arious meas ures in ABC . HJ _1_ AC Mids egment Thm. 2 HJ _1_(12) Subs titute 12 for AC . 2 HJ 6 Simplify. Class When s olv ing a problem, it is s ometimes nec es s ary to c ons ider more than one pos s ible c as e. It is helpful to mak e a draw ing of eac h c as e. Example Triangle Midsegment Theorem A mids egment of a triangle is parallel to a s ide of the triangle, and its length is half the length of that s ide. D ate Challenge 33 Copyright © by Holt, Rinehart and W inston. All rights reserved. � 7. What are the names of the mids egments ? Answers will vary based on _ students’ choice of letters: QR , _ _ RS, QS . 8. W hat is the mids egment triangle? Q RS Ho lt Ge o me tr y Copyright © by Holt, Rinehart and Winston. All rights reserved. 74 34 Ho lt Ge o me tr y Holt Geometry
