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Name LESSON 5-4 Date Class Reteach The Triangle Midsegment Theorem A midsegment of a triangle joins the midpoints of two sides of the triangle. Every triangle has three midsegments. $ _ _ RS is a midsegment of CDE. R is the midpoint of _ CD . S is the midpoint of CE . # 3 % _ Use the figure for Exercises 1–4. AB is a midsegment of RST. _ 1. What is_ the slope of midsegment AB and the slope of side ST ? Y 3 1; 1 ! _ _ _ _ 2. What can you conclude about AB and ST ? Since the slopes are the same, AB ST. X 4 " 2 3. Find AB and ST. AB 22 , ST 4 2 _ _ 4. Compare the lengths of AB and ST . 1 ST or ST 2AB AB __ 2 Use MNP for Exercises 5–7. _ 5. UV is a midsegment of MNP. Find the coordinates of U and V. Y - . U (1, 3), V (3, 2) _ _ 6. Show that UV MN . _ 1 and the slope of The slope of UV __ 4 _ 1. Since the slopes are the MN __ _4 _ same, UV MN. 5 6 0 X 1 MN. 7. Show that UV __ 2 1 (2 1 MN. UV 17 and MN 2 17 . Since 17 __ 17 ), UV __ 2 2 Copyright © by Holt, Rinehart and Winston. All rights reserved. 30 Holt Geometry Name LESSON 5-4 Date Class Reteach The Triangle Midsegment Theorem continued Theorem Example 0 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. 1 , . _ Given: PQ is a midsegment of LMN. _ _ 1 LN Conclusion: PQ LN , PQ __ 2 " You can use the Triangle Midsegment Theorem to find various measures in ABC. 1 AC HJ __ Midsegment Thm. 2 1 (12) HJ __ Substitute 12 for AC. 2 HJ 6 Simplify. 1 AB JK __ 2 __ 4 1 AB 2 8 AB _ ( * ! + # _ Midsegment Thm. HJ || AC Midsegment Thm. Substitute 4 for JK. mBCA mBJH Corr. Thm. Simplify. mBCA 35° Substitute 35° for mBJH. Find each measure. 8. VX 23 9. HJ 54 ( 8 7 92° 10. mVXJ 11. XJ 13. DE * 6 27 Find each measure. 12. ST ' 3 % 72 $ 22 14. mDES 48° 15. mRCD 48° Copyright © by Holt, Rinehart and Winston. All rights reserved. 4 # 2 31 Holt Geometry Name LESSON 5-4 Date Class Name Practice A LESSON 5-4 The Triangle Midsegment Theorem Use the Triangle Midsegment Theorem to name parts of the figure for Exercises 1–5. _ � � _ 3. a segment that has the same length as BD _ 4. a segment that has half the length of AC _ � � ������ � � 6. Use the Midpoint Formula to find the coordinates of G. ( 7. Use the Midpoint Formula to find the coordinates of H. ( _ 0 3 , 10. If two segments have same slope, then the segments _ the_ are parallel. Are DF and GH parallel? � ��� 12 16. PU 12 18. m�SUR Name LESSON 5-4 22 19. m�PRQ 55° 1522.5 mi Date _ Holt Geometry Class Reasons 1. Given 2. Midsegment Theorem 3. Definition of perimeter 4. Substitution 28 Copyright © by Holt, Rinehart and Winston. All rights reserved. Name 5-4 Date Class Holt Geometry Reteach The Triangle Midsegment Theorem A midsegment of a triangle joins the midpoints of two sides of the triangle. Every triangle has three midsegments. � � (0, 2�) � � � � � �(0, 0) 2ab � � 4. The perimeter of �STU � _1_ PQ � _1_ QR 2 2 � _1_ RP. _ _ � (2�, 0) RS is a midsegment of �CDE. R is the midpoint of _ CD. S is the midpoint of CE. D (0, b), E (a, b), F (a, 0) � 3. Pedro knows it will be easy to find the area of �EFD if �DEF is a right angle. Write a proof that �DEF � �A. _ 4. � 3. The perimeter of �STU � ST � TU � US. LESSON Pedro has a hunch about the area of midsegment triangles. He 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. � � 2 The Triangle Midsegment Theorem 2. Find the coordinates of the midpoints D, E, and F. � 5. The perimeter �STU � _1_ (PQ � QR � RP) 5. Distributive Property 2 of � Practice C 1. Find the area of �ABC. 3045 mi Possible answer: � 55° 27 Copyright © by Holt, Rinehart and Winston. All rights reserved. San Juan 8. Find the perimeter of the midsegment triangle within the Bermuda Triangle. _ _ � 17. m�SUP Bermuda Miami 7. Use the distances in the chart to find the perimeter of the Bermuda Triangle. Statements _ _ _ 1. US, ST, and TU are midsegments of �PQR. 2. ST � _1_ PQ, TU � _1_ QR, US � _1_ RP 2 2 2 15. QR 125° � Dist. (mi) Miami to San Juan 1038 Miami to Bermuda 1042 Bermuda to San Juan � 965 10. Given: US , ST , and TU are midsegments of �PQR. Prove: The perimeter of �STU � _1_(PQ � QR � RP). 2 � 14. ST � � It is half the perimeter of the Bermuda Triangle. �� �� � 58° 6. m�D 9. How does the perimeter of the midsegment triangle compare to the perimeter of the Bermuda Triangle? � � 58° 58° 4. m�HIF 122° 18.2 � 17.5 � Write a two-column proof that the perimeter of a midsegment triangle is half the perimeter of the triangle. yes 6 3 yes 11. Use the Distance Formula to find DF. Use the Triangle Midsegment Theorem and the figure for Exercises 14–19. Find each measure. ) ) 0 0 _ 9. Use the Slope Formula to find the slope of GH. 12. Use the Distance Formula to find GH. 13. Does GH � _1_ DF ? 2 2 2 , 8. Use the Slope Formula to find the slope of DF. 9.1 � 35 2. DF The Bermuda Triangle is a region in the Atlantic Ocean off the southeast coast of the United States. The triangle is bounded by Miami, Florida; San Juan, Puerto Rico; and Bermuda. In the figure, the dotted lines are midsegments. � ������� � 9.1 5. m�HGD � ������ � The Triangle Midsegment Theorem 3. GE BC 5. a segment that has twice the length of EC � Practice B 1. HI _ DE _ DE _ AD _ DE _ � � Complete Exercises _6–13 to show that _ midsegment GH is parallel to DF and that GH � _1_ DF. 2 Class Use the figure for Exercises 1–6. Find each measure. � 1. a midsegment of �ABC 2. a segment parallel to AC Date � � _ 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 _and �AFD are alternate interior angles, so �EDF _�EDF � �AFD. DF � DF by the Reflexive Property, thus �EFD � �ADF. By CPCTC, �DEF � �A. _1_ ab 2 Find the area of �EFD. Use the figure for Exercises 1–4. AB is a midsegment of �RST. _ 1. What is_ the slope of midsegment AB and the slope of side ST ? � � ������ � �1; �1 _ ������� _ _ _ � 2. What can you conclude about AB and ST ? Since the slopes are the same, AB � ST. � � � ������� �� � ������� �������� 5. Compare the areas of �ABC and �EFD. 3. Find AB and ST. � �ABC has four times the area of �EFD. � AB � 2�2 , ST � 4�2 _1_ ab 6. Pedro has already shown that �EFD � �ADF. Calculate the area of �ADF. _ _ 4. Compare the lengths of AB and ST . 2 AB � _1_ ST or ST � 2AB 7. Write a conjecture about congruent triangles and area. 2 Possible answer: Congruent triangles have equal area. Use �MNP for Exercises 5–7. _ Pedro already knows some things about the area of the midsegment triangle of a right triangle. But he thinks he can expand his theorem. Before he can get to that, however, he has to show another property of triangles and area. 5. UV is a midsegment of �MNP. Find the coordinates of U and V. � U (�1, 3), V (3, 2) � _ � ������ � _ _ 6. Show that UV � MN . 8. Find the area of �WXY, �WXZ, and �YXZ. � 16; 6; 10 � �������� � � � The slope of UV � � _1_ and the slope of 4 _ MN � � _1_. Since the slopes are the _4 _ same, UV � MN. � 9. Compare the total of the areas of �WXZ and �YXZ to the area of �WXY. Possible answer: The total of the areas of �WXZ and �YXZ is equal to the area of �WXY. 7. Show that UV � _1_ MN. � 2 10. Write a conjecture about the areas of triangles within a larger triangle. � � �� � � � ������� � UV � � 17 and MN � 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 Winston. All rights reserved. 29 Copyright © by Holt, Rinehart and Winston. All rights reserved. Holt Geometry Copyright © by Holt, Rinehart and Winston. All rights reserved. 73 30 Holt Geometry Holt Geometry Name LESSON 5-4 Date Class Name Reteach LESSON The Triangle Midsegment Theorem 5-4 continued Theorem � � _ Midsegment Thm. Substitute 4 for JK. m�BCA � m�BJH Corr. � Thm. Simplify. m�BCA � 35° Substitute 35° for m�BJH. 9. HJ � 54 case 1: EF � FG � 7 and EG � 8; case 2: EF � FG � 8 and EG � 6 �� 92° � Find each measure. 12. ST � 72 13. DE � 22 � � � ��� �� 14. m�DES � 48° 15. m�RCD � 48° Two cases; the midsegment joins If the midsegment joins the sides the sides with lengths 12 and 18. with lengths 12 and 18, then the The midsegment joins the side third side is 18. If the midsegment with lengths 12 and x. joins the side with lengths 12 and x, then it is impossible to find the � length of the third side. �� � 31 Copyright © by Holt, Rinehart and Winston. All rights reserved. Name Date Class Holt Geometry Problem Solving LESSON 1. The vertices of �JKL are J(�9, 2), K(10, 1), _ parallel and_L (5, 6). CD is the midsegment _ to JK. What is the length of CD ? Round to the nearest tenth. 5-4 2. In �QRS, QR � 2x � 5, RS � 3x � 1, and SQ � 5x. What is the perimeter of the midsegment triangle of �QRS ? 9.5 � 9.2 mi � _ 3. Which midsegment is parallel to side QS ? _ MN 4 mi � 1.5 mi � � _ 5. LN � F 150� G 140� _ � MR � � � H 110� J 30� � � _ 8. The measure of �AYW is 50�. What is the measure of �VWB ? Copyright © by Holt, Rinehart and Winston. All rights reserved. _ QM 6 cm 6. Draw the midsegments in �ABC. 6. In triangle HJK, m�H � 110�, m�J � 30°, and m�K � 40�. If R is the midpoint of _ _ JK, and S is the midpoint of HK, what is m�JRS ? the midpoint of AB, On the balance beam, V is _ and W is the midpoint of YB. _ 7. The length of VW is 1 _7_ feet. What is AY ? 8 A _7_ ft C 3 _3_ ft 8 4 15 ft D 7 _1_ ft B ___ 16 2 F 45� G 50� _ _ 4. If side RS is 12 cm, how long is LM? 1.7 mi Choose the best answer. Use the diagram for Exercises 7 and 8. � � The 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. �(12, �3) � C 240 cm2 D 480 cm2 � �LMN � 8 � (1, �1) 4. The diagram at right shows horseback riding trails. Point B _ is the halfway point _along path AC. Point _D is the _halfway point along path CE. The paths along BD and AE are parallel. If riders travel from A to B to D to E, and then back to A, how far do they travel? � _ _ _ 2. What is the midsegment triangle in �QRS ? 0 5. Right triangle FGH has midsegments of length 10 centimeters, 24 centimeters, and 26 centimeters. What is the area of �FGH ? � 1. Name the midsegments in �QRS. LM; MN; NL Yes; X is the midpoint of LN, and 2 Identify Relationships �(4, 8) _ Holt Geometry A midsegment triangle is formed from the three midsegments of a triangle. 6 Y is the midpoint of ML . Class Reading Strategies � _ Date A midsegment of a triangle is a segment that joins the midpoints of two sides of the triangle. 5x � 2 3. Is XY a midsegment of �LMN if its endpoints are X(8, 2.5) and Y(6.5, �2)? Explain. 32 Copyright © by Holt, Rinehart and Winston. All rights reserved. Name The Triangle Midsegment Theorem A 60 cm B 120 cm2 4. Find the length of the third side of �ABC by considering both cases. � � � � � 3. How many cases are there to consider when making a conclusion about the third side of the triangle? Explain. � � 18 12 � 27 11. XJ � � Use �ABC for Exercises 4 and 5. A midsegment of the triangle is 9. ��� �� � � 2. Find the lengths of the triangle’s sides for each of the cases in Exercise 1. � 10. m�VXJ � 5-4 � � HJ || AC 23 � � � � �� � Midsegment Thm. 8. VX � � � � � � � � � ��� Find each measure. LESSON � � � � _ midsegment connects Case 2: The the base EG and one of the congruent sides of �EFG. Case 1: The midsegment connects the two congruent sides EF and FG. � _ _ Consider Different Cases 1. Describe two possible cases and make a drawing of each. � Given: PQ is a midsegment of �LMN. _ _ Conclusion: PQ � LN , PQ � _1_ LN 2 JK � _1_ AB 2 4 � _1_ AB 2 8 � AB Challenge Triangle EFG is an isosceles triangle _ with EF � FG and with the perimeter equal to 22 units. A midsegment, QR , of �EFG is equal to 4 units. � You can use the Triangle Midsegment Theorem to find various measures in �ABC. HJ � _1_ AC � Midsegment Thm. 2 HJ � _1_ (12) Substitute 12 for AC. 2 Simplify. HJ � 6 Class When solving a problem, it is sometimes necessary to consider more than one possible case. It is helpful to make a drawing of each case. 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. Date � � 7. What are the names of the midsegments? Answers will vary based on � _ students’ choice of letters: QR, _ _ � RS, QS. � � 8. What is the midsegment triangle? H 90� J 130� �QRS 33 Copyright © by Holt, Rinehart and Winston. All rights reserved. Holt Geometry Copyright © by Holt, Rinehart and Winston. All rights reserved. 74 34 Holt Geometry Holt Geometry