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5-4

5-4. Midsegment Theorem. Warm Up. Lesson Presentation. Lesson Quiz. Holt Geometry. 5.4 Midsegment Theorem. Warm Up Use the points A (2, 2), B (12, 2) and C (4, 8) for Exercises 1–5. 1. Find X and Y , the midpoints of AC and CB . 2. Find XY . 3. Find AB .

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5-4

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  1. 5-4 Midsegment Theorem Warm Up Lesson Presentation Lesson Quiz Holt Geometry

  2. 5.4 Midsegment Theorem Warm Up Use the points A(2, 2), B(12, 2) and C(4, 8) for Exercises 1–5. 1.Find X and Y, the midpoints of AC and CB. 2. Find XY. 3. Find AB. 4. Find the slope of AB. 5. Find the slope of XY. 6. What is the slope of a line parallel to 3x + 2y = 12? (3, 5), (8, 5) 5 10 0 0

  3. 5.4 Midsegment Theorem Objective Prove and use properties of triangle midsegments.

  4. 5.4 Midsegment Theorem Vocabulary midsegment of a triangle

  5. 5.4 Midsegment Theorem A midsegment of a triangleis a segment that joins the midpoints of two sides of the triangle. Every triangle has three midsegments, which form the midsegment triangle.

  6. 5.4 Midsegment Theorem The vertices of ∆XYZ are X(–1, 8), Y(9, 2), and Z(3, –4). M and N are the midpoints of XZ and YZ. Show that and . Example 1: Examining Midsegments in the Coordinate Plane Step 1 Find the coordinates of M and N.

  7. 5.4 Midsegment Theorem Step 2 Compare the slopes of MN and XY. Since the slopes are the same, Example 1 Continued

  8. 5.4 Midsegment Theorem Step 3 Compare the heights of MN and XY. Example 1 Continued

  9. 5.4 Midsegment Theorem The vertices of ΔRST are R(–7, 0), S(–3, 6), and T(9, 2). M is the midpoint of RT, and N is the midpoint of ST. Show that and Check It Out! Example 1 Step 1 Find the coordinates of M and N.

  10. 5.4 Midsegment Theorem Step 2 Compare the slopes of MN and RS. Since the slopes are equal . Check It Out! Example 1 Continued

  11. 5.4 Midsegment Theorem Step 3 Compare the heights of MN and RS. The length of MN is half the length of RS. Check It Out! Example 1 Continued

  12. 5.4 Midsegment Theorem The relationship shown in Example 1 is true for the three midsegments of every triangle.

  13. 5.4 Midsegment Theorem Example 2A: Using the Triangle Midsegment Theorem Find each measure. BD ∆ Midsegment Thm. Substitute 17 for AE. BD = 8.5 Simplify.

  14. 5.4 Midsegment Theorem Example 2B: Using the Triangle Midsegment Theorem Find each measure. mCBD ∆ Midsegment Thm. mCBD = mBDF Alt. Int. s Thm. mCBD = 26° Substitute 26° for mBDF.

  15. 5.4 Midsegment Theorem Check It Out! Example 2a Find each measure. JL ∆ Midsegment Thm. Substitute 36 for PN and multiply both sides by 2. 2(36) = JL 72 = JL Simplify.

  16. 5.4 Midsegment Theorem Check It Out! Example 2b Find each measure. PM ∆ Midsegment Thm. Substitute 97 for LK. PM = 48.5 Simplify.

  17. 5.4 Midsegment Theorem Check It Out! Example 2c Find each measure. mMLK ∆ Midsegment Thm. mMLK = mJMP Similar triangles mMLK = 102° Substitute.

  18. 5.4 Midsegment Theorem Example 3: Indirect Measurement Application In an A-frame support, the distance PQ is 46 inches. What is the length of the support ST if S and T are at the midpoints of the sides? ∆ Midsegment Thm. Substitute 46 for PQ. ST= 23 Simplify. The length of the support ST is 23 inches.

  19. 5.4 Midsegment Theorem Check It Out! Example 3 What if…? Suppose Anna’s result in Example 3 (p. 323) is correct. To check it, she measures a second triangle. How many meters will she measure between H and F? ∆ Midsegment Thm. Substitute 1550 for AE. HF= 775 m Simplify.

  20. 5.4 Midsegment Theorem Lesson Quiz: Part I Use the diagram for Items 1–3. Find each measure. 1.ED 2.AB 3. mBFE 10 14 44°

  21. 5.4 Midsegment Theorem Lesson Quiz: Part II 4. Find the value of n. 5.∆XYZ is the midsegment triangle of∆WUV. What is the perimeter of∆XYZ? 16 11.5

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