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 LESSON8 . 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 © Houghton Mifflin Houghton © Company Harcourt Publishing B C PERFORMANCE TASK B C View the Engage section online. Discuss the photo, 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 join the midpoints of a triangle’s sides related 8 . 4 Midsegments of Triangles Resource Locker Essential Question: toHow the are triangle’s the segments sides? that HARDCOVER PAGES 339346 . 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 08L4.indd 395 GE_MNLESE385795_U2M 395 Lesson 8 . 4 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” DO NOT EDIT--Changes must be made through “File info” CorrectionKey=NL-A;CA-A CorrectionKey=NL-A;CA-A 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 y1 y2 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 Mifflin Houghton © Company Harcourt Publishing _ 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 - y1 _ _ 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. Midsegments of Triangles 396 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-B;CA-B y B The vertices of △LMN are L 2, 7 , M 10, 9 , and N 8, 1 . P is the 10 _ ( ) ( _) ( ) 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 .