Midsegments of Triangles of Triangles Essential Question: How Are the Segments That Join the Midpoints of a Triangle’S Sides Related to the Triangle’S Sides?

Home , Line segment, Midpoint

DO NOT EDIT--Changes must be made through “File info” CorrectionKey=NL-C;CA-C DO NOT EDIT--Changes must be made through “File info” CorrectionKey=NL-A;CA-A LESSON 8 . 4 Name Class Date Midsegments 8.4 Midsegments of Triangles of Triangles Essential Question: How are the segments that join the midpoints of a triangle’s sides related to the triangle’s sides?

Resource Common Core Math Standards Locker The student is expected to: Explore Investigating Midsegments of a Triangle COMMON CORE G-CO.C.10 The midsegment of a triangle is a line segment that connects the midpoints of two sides of the triangle. Every triangle has three midsegments. Midsegments are often used to add Prove theorems about triangles. Also G-CO.D.12, G-GPE.B.4, G-GPE.B.5 _ rigidity to structures. In the support for the garden swing shown, the crossbar DE is a Mathematical Practices midsegment of △ABC

COMMON CORE MP.2 Reasoning Language Objective Explain to a partner why a drawn segment in a triangle is or is not a midsegment.

ENGAGE You can use a compass and straightedge to construct the midsegments of a triangle. Essential Question: How are the A Sketch a scalene triangle and label B Use a compass to find the midpoint the vertices A, B, and C. of AB ¯ . Label the midpoint D. segments that join the midpoints of a A A triangle’s sides related to the triangle’s D sides? B C B C Each midsegment is half the length of the side to which it is parallel; the sum of the lengths of the _ _ midsegments is half the perimeter of the triangle. C Use a compass to find the midpoint of AC¯ . Label D Use a straightedge to draw DE . DE is one of the the midpoint E. midsegments of the triangle. A A

D E D E PREVIEW: LESSON

drawing attention to the horizontal crossbeams that Repeat the process to find the other two midsegments of △ABC. You may want to E _ extend between the slanting beams and stabilize label the midpoint of BC as F. them. Then preview the Lesson Performance Task. Module 8 through “File info” 395 Lesson 4 DO NOT EDIT--Changes must be made CorrectionKey=NL-A;CA-A Date Class

Name

8. 4 Midsegments of Trianglesjoin the midpoints of a triangle’s sides related Resource Locker Essential Question: toHow the are triangle’s the segments sides? that HARDCOVER PAGES 339346 . Also G-CO.D.12, G-GPE.B.4, G-GPE.B.5

COMMON CORE G-CO.C.10 Prove theorems about triangles

Investigating Midsegmentsects the midpoints of ofa two Triangle sides of Explore _ Midsegments are often usedDE to isadd a of a triangle is a line segment that conn The midsegment swing shown, the crossbar GE_MNLESE385795_U2M08L4.indd 395 the triangle. Every triangle has three midsegments. 02/04/14 2:30 AM rigidity to structures. In the support for the garden midsegment of △ABC Turn to these pages to find this lesson in the ents of a triangle.

ghtedge to construct the midsegm midpoint Use a compass to find the You can use a compass and strai  d label of AB¯ . Label the midpoint D. hardcover student A  Sketch a scalene Btriangle, and C an. the vertices A, D A C B C B edition. _ _ is one of the DE . DE Use a straightedge to draw AC¯ . Label  idpoint of midsegments of the triangle. A  Use a compass to find the m E the midpoint E. D C A B E D C

B © Houghton Mifflin Harcourt Publishing Company Publishing Harcourt Mifflin Houghton ©

△ABC. You may want to

the other two midsegments of Lesson 4 _  Repeat the process to findBC as F. label the midpoint of 395

02/04/14 2:31 AM Module 8

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Reflect _ _ 1. Use a ruler to compare the length of DE to the length of BC . What does this tell you EXPLORE _ _ about DE and BC ? _ _ The length of DE is half the length of BC . Investigating Midsegments _ 2. Use a protractor to compare m ADE and m ABC. What does this tell you about DE _ ∠ ∠ of a Triangle and BC ? Explain. _ _ m∠ADE = m∠ABC, so DE ∥ BC since corresponding angles are congruent. 3. Compare your results with your class. Then state a conjecture about a midsegment of INTEGRATE TECHNOLOGY a triangle. A midsegment of a triangle is parallel to the third side of the triangle and is half as long as Students have the option of completing the

the third side. midsegments activity either in the book or online.

Explain 1 Describing Midsegments on a Coordinate Grid QUESTIONING STRATEGIES

You can confirm your conjecture about midsegments using the formulas for the midpoint, How are the length of the midsegment and the slope, and distance. third side of the triangle related? The Example 1 Show that the given midsegment of the triangle is parallel to the third side midsegment is one-half the length of the third side. of the triangle and is half as long as the third side.

 The vertices of △GHI are G( -7, -1) , H( -5, 5) , and I (1, 3) . J is y _ _ _ 6 the midpoint of GH , and K is the midpoint of IH . Show that JK ∥ H _ 1 _ GI and JK __ GI. Sketch JK . = 2 K 4 EXPLAIN 1 I x x y 1 y 2 Step 1 Use the midpoint formula, _1 + 2 , _ + , to find the 2 2 J coordinates of J and K. Describing Midsegments on a ( ) x _ 1 5 8 4 2 0 2 Coordinate Grid The midpoint of GH is _-7 - 5 , _- + 6, 2 . - G - - 2 2 = ( - ) Graph and label this point( J. ) -2 Publishing Harcourt Company © Houghton Mifflin

_ 5 1 5 3 4 The midpoint of IH is _- + , _+ 2, 4 . Graph - INTEGRATE MATHEMATICAL 2 2 = ( - ) ( ) _ and label this point K. Use a straightedge to draw JK . PRACTICES Focus on Math Connections _ y2 - y 1 _ _ Step 2 Use to compare the slopes of JK and GI . ( x 2 - x 1 ) _ _4 2 _1 _ _3 - ( -1) _1 MP.1 Remind students that two lines with equal Slope of JK = - = Slope of GI = = -2 - ( -6) 2 1 - ( -7) 2 _ _ slopes are parallel and that the slope of a segment is Since the slopes are the same, JK ∥ GI . ____ the difference of its y-coordinates divided by the 2 2 _ _ Step 3 Use ( x 2 - x 1) + y 2 - y 1 to compare the lengths of JK and GI . √ ____( ) difference of its x-coordinates. Also remind them to 2 2 _ _ JK = √( -2 - ( - 6) ) + (4 - 2) = √ 20 = 2√ 5 subtract the coordinates in the same order. ――――――――――2 2 _ _ GI = √( 1 - ( -7 )) + ( 3 - ( -1)) = √ 80 = 4√ 5 _ 1 _ 1 Since 2 5 _ 4 5 , JK _ GI. √ = 2 ( √ ) = 2

Module 8 396 Lesson 4

PROFESSIONAL DEVELOPMENT

GE_MNLESE385795_U2M08L4.indd 396 3/27/14 10:21 AM Math Background A midsegment of a triangle is a segment that joins the midpoints of two sides of the triangle. Together, the three midsegments of a triangle form the sides of the midsegment triangle. Using the Triangle Midsegment Theorem and the SSS Triangle Congruence Theorem, it can be proven that the four small triangles formed by the midsegments are congruent. Since the four triangles together form the original triangle, each small triangle has one-fourth of its area.

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y The vertices of △LMN are L 2, 7 , M 10, 9 , and N 8, 1 . P is the 10 B _ ( ) ( _) ( ) M QUESTIONING STRATEGIES midpoint of LM , and Q is the midpoint of MN . P 8 _ _ 1 _ Show that PQ LN and PQ _ LN . Sketch PQ . L To which side of the triangle does the ∥ = 2 6 midsegment appear to be parallel? the side 10 9 2 7 6 Q Step 1 The midpoint of LM_ _+ , _+ , 8 . 4 that does not contain the endpoints of the = 2 2 = Graph and label this point P. ( ) midsegment 2 8 10 1 9 _ __ + __ + N x The midpoint of NM = , How do you find the midsegments of a ( 2 2 ) 0 2 4 6 8 10 triangle in the coordinate plane? Use the 9 5 _ = , . Graph and label this point Q. Use a straightedge to draw PQ . Midpoint Formula to find the coordinates of the ( ) 1 - 7 _ _5 - 8 _ __ midpoint of two sides of the triangle. Plot the Step 2 Slope of PQ = = -1 Slope of LN = = -1 9 6 8 - 2 midpoints and connect them to form a midsegment. - _ _ parallel What are some ways you can show that two Since the slopes are the same, PQ and LN are . ______2 line segments are parallel when using the 2 _ _ Step 3 PQ 9 6 ( 5 8 ) 9 9 18 3√ 2 Triangle Midsegment Theorem in a coordinate = - + - = √ + = √ = _____√( ) ___ _ _ 2 plane? Sample answer: Show that the lines have the 2 LN = 8 - 2 + 1 - 7 = 36 + 36 = 72 = 6 2 same slope. √ ( ) ( ) √ √ √ _ _ 1 1 Since 3 2 _ 6 2 PQ ― _ LN ― √ = 2( √ ) , = 2 . _ half _ The length of PQ is the length of LN .

Your Turn

4. The vertices of XYZ are X 3, 7 , Y 9, 11 , and Z 7, 1 . U is the y _△ ( ) ( )_ ( ) Y midpoint of XY , and W is the midpoint of XZ . Show that _ _ 1 _ 10 UW ∥ YZ and UW _ YZ. Sketch XYZ and UW . U = 2 △ 3 9 7 11 U _+ , _+ 6, 9 ; 8 = 2 2 = ( ) X ( 3 7 7 1 ) 6 W _+ , _+ = 2 2 ( ) 4 = (5, 4) ; W 2 _4 - 9 _1 11 slope of UW― = = 5; slope of YZ― = - = 5; 5 - 6 7 - 9 Z x 0 2 4 6 8 10 Since the slopes are the same, UW― ∥ YZ― . © Houghton Mifflin Houghton © Company Harcourt Publishing 2 2 UW = √( ―――――――5 - 6) + (4 - 9) = √―26 ; 2 2 YZ = √(――――――― 7 - 9) + (1 - 11) = 2 √―26 ; S0 UW _1 YZ. = 2

Module 8 397 Lesson 4

COLLABORATIVE LEARNING

GE_MNLESE385795_U2M08L4.indd 397 5/22/14 10:14 PM Small Group Activity Have one student in each group use geometry software to construct the midsegment of a triangle. Ask another student in the group to measure the midsegment and the sides of the triangle and make a conjecture about the relationship of the two measures. Ask a third student to measure the angles to make a conjecture about midsegment and the third side of the triangle. Have the fourth student drag the vertices of the triangle to verify the conjectures made by the other students: the midsegments are half the measure of the third side and are parallel to the third side.

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Explain 2 Using the Triangle Midsegment Theorem EXPLAIN 2 The relationship you have been exploring is true for the three midsegments of every triangle. Using the Triangle Triangle Midsegment Theorem Midsegment Theorem The segment joining the midpoints of two sides of a triangle is parallel to the third side, and its length is half the length of that side.

You explored this theorem in Example 1 and will be proving it later in this course. INTEGRATE MATHEMATICAL PRACTICES Example 2 Use triangle RST. Focus on Math Connections  Find UW. S MP.1 Point out that if the midsegments of a triangle By the Triangle Midsegment Theorem, the length of are found, then the Triangle Midsegment Theorem _ _ midsegment UW is half the length of ST . can be used to state that the midsegment is half the _1 U 5.6 UW = ST 10.2 2 41° V length of the third side and is parallel to the third UW _1 10.2 = 2( ) side. Explain that the theorem gives a way to R UW = 5.1 indirectly find the side lengths of a triangle, and W T that it gives similar triangles, which will be studied  Complete the reasoning to find m ∠SVU. in Unit 4. Midsegment Thm. _ _ △ UW ǁ ST Alt. Int. Angles Thm. m ∠SVU = m ∠VUW 41 m VUW 41 QUESTIONING STRATEGIES Substitute ° for ∠ m ∠SVU = ° In the example, how do you know that you can Reflect use the Triangle Midsegment Theorem? The © Houghton Mifflin Harcourt Publishing Harcourt Company © Houghton Mifflin 5. How do you know to which side of a triangle a midsegment is parallel? midsegment of a triangle was found, so the Triangle Since a midsegment connects the midpoints of two sides, it is parallel to the third side. Midsegment Theorem applies.

Your Turn P K J 6. Find JL, PM, and m ∠MLK. 1 1 39 AVOID COMMON ERRORS JL 2 PN 2 39 78; PM _ KL _ 95 47 5; 105° N = ( ) = ( ) = = 2 = 2 ( ) = . M 95 m MLK m JMP 105 Sometimes students write the incorrect algebraic ∠ = ∠ = ° L expressions to indirectly find the lengths of segments Elaborate in triangles when using the Triangle Midsegment 6 6 Theorem. Have these students use the theorem to 7. Discussion Explain why XY ¯ is NOT a midsegment of the triangle. The endpoints of XY¯ are not midpoints of the sides of X Y 5 5 write the relationships among the segments before the triangle. they substitute algebraic expressions for the segments. The correct expressions should give the

Module 8 398 Lesson 4 correct equations to solve for the lengths.

DIFFERENTIATE INSTRUCTION GE_MNLESE385795_U2M08L4Modeling 398 6/10/15 12:22 AM Have students cut a large scalene triangle from heavy paper. Have them construct the midpoints of two sides and connect them, forming a midsegment. Then have them cut along this segment to form a triangle and a trapezoid. Have them rotate the small triangle and place it next to the trapezoid to form a parallelogram. Ask students to make conjectures about the relationship between the midsegment and the base of the triangle.

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8. Essential Question Check–In Explain how the perimeter of △DEF ELABORATE compares to that of △ABC. A QUESTIONING STRATEGIES F D How could you use a property of midsegments 1 The perimeter of △DEF = _ perimeter of △ABC . C B 2 ( ) to find a measurement for something that is E difficult to measure directly? Sample answer: If you cannot measure directly across something like a pond, you can construct a triangle around it so that Evaluate: Homework and Practice

the distance you want to measure is a midsegment • Online Homework 1. • Hints and Help of the triangle. Then you can measure the third side Use a compass and a ruler or geometry software to construct an obtuse triangle. • Extra Practice of the triangle and divide it by 2 to find the measure Label the vertices. Choose two sides and construct the midpoint of each side; then label and draw the midsegment. Describe the relationship between the length of the you are looking for. midsegment and the length of the third side. Drawings will vary. Students should conclude that the midsegment is half the length of the third side. SUMMARIZE THE LESSON 2. The vertices of △WXY are W -4, 1 , X( 0, -5 ) , and Y 4, -1 . A is the midpoint of _ (_ ) _ _ ( ) WY , and B is the midpoint of XY . Show that AB WX and AB _ 1 WX. Have students fill out a chart to summarize ǁ = 2 _ _ what they know about triangle The coordinates of A and B are (0, 0) and (2, -3). The slope of AB and WX 3 _  Sample: is , so the lines are parallel. The length of AB is 13 and the length of midsegments. - _2 ― _ 1  WX is 2 13 , so AB = _ WX. Triangle Midsegment ― 2 3. The vertices of △FGH are F -1, 1 , G( -5, 4 ) , and H -5, –2 . X is the midpoint of A line segment that _ ( _ ) _ _ ( ) 1 Definition __ FG, and Y is the midpoint of FH . Show that XY ǁGH and XY = 2 GH. connects the 5 1 _ The coordinates of X and Y are 3, _ and 3, _ . The slope of XY midpoints of two - 2 - - 2 and GH_ is undefined, so the lines( are parallel.) ( The length) of XY_ is 3 and sides of the triangle _ 1 the length of GH is 6, so XY _ GH. Properties Half the length of the = 2 _ third side; parallel to 4. One of the vertices of △PQR is P( 3, –2) . The midpoint of PQ is M( 4, 0) . _ _ _ The midpoint of QR is N 7, 1 . Show that MN PR and MN _ 1 PR. the third side ( ) ǁ = 2 3 x 2 y _+ 4, So x 5; _- + 0, So y 2. Example Answers will vary. 2 = = 2 = = The coordinates of Q are 5, 2 . Non-example Answers will vary. ( ) 5 x 2 + y © Houghton Mifflin Houghton © Company Harcourt Publishing _+ 7, So x 9; _ 1, So y 0. 2 = = 2 = = The coordinates of R are (9, 0) . 1 The slope of MN and PR is _ , so the lines are parallel. The length of MN is ¯ ¯ 3 ¯ _ 1 10 and the length of PR is 2 10 , so MN _ PR. √ ¯ √― = 2

Module 8 399 Lesson 4

LANGUAGE SUPPORT GE_MNLESE385795_U2M08L4.inddCommunicate 399 Math 3/22/14 2:36 PM Divide students into pairs. Provide each pair with pictures of triangles with segments joining two sides. Have one student explain why a drawing does or does not show a midsegment. Encourage the use of the words parallel and midpoint. Have students measure the distance between the segment and the base at different intervals to see if the segment and the base are parallel, and to measure the sides joined by the segment, to see if the segment’s endpoints are the midpoints of the sides. Students switch roles and repeat the process.

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_ 5. One of the vertices of ABC is A 0, 0 . The midpoint of AC is J _3 , 2 . The midpoint _ △ _ _( ) 2 of BC is K 4, 2 . Show that JK BA and JK _ 1 BA. ( ) ( ) ǁ = 2 0 x 3 0 + y + _ _ 2 So, the coordinates of C are 3, 4 . EVALUATE _2 = 2 2 = ( ) x = 3 y = 4 Use the midpoint formula to find the coordinates of B (x, y) . 3 x 4 + y _+ 4 _ 2 2 = 2 = 3 + x = 8 4 + y = 4 x = 5 y = 0 So, the coordinates of B are (5, 0) . ASSIGNMENT GUIDE 5 The slope of JK and BA is 0, so the lines are parallel. The length of JK is _ ¯ ¯ ¯ 2 _ 1 Concepts and Skills Practice and the length of BA is 5, so JK = _ BA. 2 Explore Exercises 1–3 Find each measure. A Investigating Midsegments of a Triangle X Y 68° Example 1 Exercises 4–5 4.6 Describing Midsegments on a B C Z Coordinate Grid 15.8 Example 2 Exercises 6–15 6. XY 7. BZ 1 1 1 1 Using the Triangle Midsegment XY _ BC _ 15 8 7 9 BZ _ BC _ 15 8 7 9 = 2 = 2 ( . ) = . = 2 = 2 ( . ) = . Theorem

INTEGRATE MATHEMATICAL PRACTICES 8. AX 9. YZC 1 m∠ Since AX _ AB, find AB first. YZC and XYZ are alternate interior Focus on Math Connections = 2 ∠ ∠ Publishing Harcourt Company © Houghton Mifflin angles, so they are congruent. AB = 2 (YZ) = 2 (4.6) = 9.2 MP.1 Remind students of the key concepts they m YZC 68° 1 1 ∠ = AX _ AB _ 9 2 4 6 = 2 = 2 ( . ) = . need to do the exercises. Define midsegment. Make a sketch on the board showing students how to draw a midsegment, and point out that every triangle has three midsegments. Add that, in this lesson, students 10. m∠BXY m∠BXY + m∠ZYX = 180° (same-side interior angles) will primarily work with one midsegment at a time. m∠BXY + 68° = 180° m∠BXY = 112°

Module 8 400 Lesson 4

COMMON GE_MNLESE385795_U2M08L4Exercise 400 Depth of Knowledge (D.O.K.) CORE Mathematical Practices 10/14/14 9:19 PM

1–15 2 Skills/Concepts MP.6 Precision 16–21 2 Skills/Concepts MP.2 Reasoning 22 3 Strategic Thinking MP.4 Modeling 23 3 Strategic Thinking MP.6 Precision 24 3 Strategic Thinking MP.4 Modeling

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Algebra Find the value of n in each triangle. INTEGRATE MATHEMATICAL 11. 12. PRACTICES 11.3 6n 48 Focus on Communication n + 4.2 MP.3 Discuss each of the following statements as a 1 class. Have students explain how the Triangle 6n _ 48 2 (11.3) = n + 4.2 = 2( ) Midsegment Theorem justifies their responses. 22 6 n 4 2 6n = 24 . = + . 18 4 n 1. A triangle has three midsegments. n = 4 . =

2. The midsegments form a triangle with a perimeter 13. 14. 14n equal to one-half the perimeter of the original n + 12 4n + 9 triangle. 6n 3. The midsegments form a triangle with area equal to one-fourth the area of the original triangle. 6n 2 n 12 = ( + ) 14n = 2 (4n + 9) 6n 2n 24 = + 14n = 8n + 18 4n 24 = 6n = 18 n 6 = n = 3

15. Line segment XY is a midsegment of △MNP. Determine whether each of the following statements is true or false. Select the correct answer for each lettered part.

X Y

M P

a. MP = 2XY True False b. MP _1 XY True False = 2 c. MX = XN True False d. MX _ 1 NX True False = 2 e. NX = YN True False f. XY _1 MP True False = 2 © Houghton Mifflin Houghton © Company Harcourt Publishing 16. What do you know about two of the midsegments in an isosceles triangle? Explain. 1 __ Two of the midsegments will be congruent because their lengths are 2 the length of congruent sides.

Module 8 401 Lesson 4

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17. Suppose you know that the midsegments of a triangle are all 2 units long. What kind of triangle is it? PEERTOPEER DISCUSSION an equilateral triangle with sides 4 units long Ask students to discuss with a partner how to locate

_ _ _ the midsegments of a triangle drawn in the 18. In △ABC, m∠A = 80°, m∠B = 60°, m∠C = 40°. The midpoints of AB , BC , and AC are D, E, and F, respectively. Which midsegment will be the longest? Explain how you coordinate plane. Have them use the slope formula or know. distance formula and the Triangle Midsegment BC¯ is the longest side because it is opposite the largest angle, so midsegment ¯DF will be Theorem to verify that the midsegments are equal to the longest. one-half the third side and are parallel to the third 19. Draw Conclusions Carl’s Construction is building a pavilion side. Have them draw examples to justify with an A-frame roof at the local park. Carl has constructed two triangular frames for the front and back of the roof, similar to their reasoning. ABC in the diagram. The base of each frame, represented by _△ AC , is 36 feet long. He needs to insert a crossbar connecting the _ _ midpoints of AB and BC , for each frame. He has 32 feet of timber left after constructing the front and back triangles. Is this enough AVOID COMMON ERRORS to construct the crossbar for both the front and back frame? Explain. When using triangles in the coordinate plane, some

B students may assume that the Triangle Midsegment Theorem applies only to acute, scalene triangles. D E Have students work in groups to draw other types of A C triangles in the coordinate plane and use algebraic

© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Polianskyi ©Polianskyi Credits: • Image Company Publishing Harcourt Mifflin © Houghton expressions to write the triangle relationships that No. If AC_ is 36 feet long, DE_ is 18 feet long. Because he needs to insert two crossbars, he apply to these triangles. needs 2 × 18, or 36, feet of timber. He needs 4 more feet.

20. Critique Reasoning Line segment AB is a midsegment in PQR. _ △ R Kayla calculated the length of AB . Her work is shown below. Is her B answer correct? If not, explain her error. 25 2( QR) = AB Q P 2( 25) = AB A 50 = AB 1 No, it is not correct. The equation should have __ QR on the left instead of 2 QR . 2 ( )

21. Using words or diagrams, tell how to construct a midsegment using only a Igor/Shutterstock straightedge and a compass. Possible answer: First construct a triangle with a straightedge. Next use the compass to find the midpoint of two sides of the triangle. Finally, connect the two midpoints to create a midsegment parallel to the third side.

Module 8 402 Lesson 4

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JOURNAL H.O.T. Focus on Higher Order Thinking

Have students explain how to find the length of a 22. Multi–Step A city park will be shaped like a right triangle, _ and there will be two pathways for pedestrians, shown by VT midsegment when given the length of the _ and VW in the diagram. The park planner only wrote two parallel side. lengths on his sketch as shown. Based on the diagram, what will be the lengths of the two pathways?

30 yd V W

X Z T 24 yd 1 1 VW _ XZ _ 24 12 = 2 ( ) = 2 ( ) = 2 1 1 24 2 YZ 30 2 (Pyth. Thm.) VT _ YZ _ 18 9 + ( ) = = 2 ( ) = 2 ( ) = 2 576 + (YZ) = 900 VW = 12 yd; VT = 9 yd 2 ( YZ) = 324 YZ = 18

23. Communicate Mathematical Ideas △XYZ is the midsegment of △PQR. Q Write a congruence statement involving all four of the smaller triangles. What X Y is the relationship between the area of △XYZ and △PQR? P R △QXY ≅ △XPZ ≅ △YZR ≅ △ZYX; Z 1 __ area of △XYZ = 4 area of △PQR _ _ 24. Copy the diagram shown. AB is a midsegment of △XYZ. CD is a Z midsegment of △ABZ. C D a. What is the length of ¯AB ? What is the ratio of AB to XY? 1 AB _ XY , so AB is 32. The ratio of AB to XY is 32:64, or 1:2. = 2 ( ) A B b. What is the length of ¯CD ? What is the ratio of CD to XY? E F _1 CD = (AB) , so CD is 16. The ratio of CD to XY is 16:64, or 1:4. 2 X Y _ 3 64 c. __ Draw EF such that points E and F are 4 the distance from point Z to points X and Y. What is the ratio of EF to XY? What is the length _ of EF ? 3 3 EF _ XY _ 64 48 = 4 ( ) = 4 ( ) = © Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Carey Hope/iStockPhoto.com The length of ¯EF is 48, so the ratio of EF to XY is 48:64, or 3:4. d. Make a conjecture about the length of non-midsegments when compared to the length of the third side. The length of the non-midsegment compared to the length of the third side is the same as the ratio of the distance of the segment from the vertex opposite the third side compared to the whole triangle.

Module 8 403 Lesson 4

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Lesson Performance Task AVOID COMMON ERRORS The figure shows part of a common roof design using very strong and stable triangular trusses. ______Points B, C, D, F, G, and I are midpoints of AC , AE , CE , GE , HE and AH respectively. What is Students may mistakenly assume that because B, C, the total length of all the stabilizing bars inside △AEH? Explain how you found the answer. _ and D divide AE into four congruent segments, G _ A and F must divide HE into three congruent segments, _ and that J and I divide AH into three congruent B segments. Point out that because G is the midpoint of J _ 20 ft C HE , GF and FE are each only half of HG. Similarly, I 4 ft _ because I is the midpoint of AH , AJ and JI are each D only half of IH. G F H E 14 ft INTEGRATE MATHEMATICAL By the Midsegment Theorem, IC = 7 ft and CG = 10 ft because they are, respectively, half PRACTICES of HE and AH. CH = 8 ft because it is twice DG. DF = 5 ft, JB = 3.5 ft, and IB = 4 ft because they are, respectively, half of CG, IC, and HC. Focus on Critical Thinking Total length: JB IB IC CH CG DG DF 3.5 4 7 8 10 4 5 41.5 ft + + + + + + = + + + + + + = MP.3 On the roof truss diagrammed below, D, E, _ _ and F are the midpoints of, respectively, AC , AB , _ and CB . A

C B F

Compare the perimeter and area of ▵ABC with those of ▵DEF. Explain how you found your answers. Students can use the Midsegment Theorem and the _ _ _ facts that DE , EF , and DF are parallel to and half the _ _ _ length of, respectively, CB , AC , and AB to establish that (1) the perimeter of ▵DEF is half that of ▵ABC, and ( 2) the four small triangles in the diagram are congruent, which means that the area of ▵DEF is Module 8 404 Lesson 4 one-fourth that of ▵ABC. EXTENSION ACTIVITY

GE_MNLESE385795_U2M08L4.indd 404 3/22/14 2:35 PM Have students research tri-bearing roof trusses. Among topics they can report on: • the definition of tri-bearing roof truss; • the design of such a truss, together with a drawing and typical measurements; • the meaning of the term tri-bearing; • a description of the relationship between tri-bearing roof trusses and the Scoring Rubric Midsegment Theorem. 2 points: Student correctly solves the problem and explains his/her reasoning. 1 point: Student shows good understanding of the problem but does not fully solve or explain. 0 points: Student does not demonstrate understanding of the problem. Midsegments of Triangles 404