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1 1/19 Warm Up Fast answers! The altitudes are concurrent at the? Orthocenter The medians are concurrent at the? Centroid The perpendicular bisectors are concurrent at the? Circumcenter The angle bisectors are concurrent at the? Incenter January 19, and 6.3 Points of Concurrency

2 Fast answers! Which point is equidistant from the sides of a triangle? Incenter Which point is the center of balance? Centroid Which point is equidistant from the vertices? Circumcenter? January 19, and 6.3 Points of Concurrency

3 Fast answers! What point is needed to draw a circumcircle? Circumcenter What point is needed to draw an incircle? Incenter What point is needed to find the center of balance? Centroid January 19, and 6.3 Points of Concurrency

4 Do you know What a median is? How to draw a perpendicular bisector? What lines are needed to find the incenter? How to locate the circumcenter? What point is located using the medians? How to construct an altitude? Which concurrent point is the same distance from each vertex of a triangle? January 19, and 6.3 Points of Concurrency

5 Geometry 6.4 Midsegment Theorem

6 Goals Identify the midsegments of triangles. Use the midsegment properties to solve problems. January 19, 2016 Geometry 5.4 Midsegment Theorem 6

7 Construction Use a Mira and draw an acute, scalene triangle. Label the vertices A, B, and C. A C B January 19, 2016 Geometry 5.4 Midsegment Theorem 7

8 Construction Use a Mira and find the midpoint of AC. Label it R. Find the midpoint of CB and label it S. Draw RS. A R C S B January 19, 2016 Geometry 5.4 Midsegment Theorem 8

9 Midsegment A midsegment of a triangle is the segment connecting the midpoints of two sides. RS is a midsegment of ABC. A triangle has three midsegments. R C S A B In some texts, the midsegment is called the midline. January 19, 2016 Geometry 5.4 Midsegment Theorem 9

10 Midsegment Properties A midsegment of a triangle is parallel to the third side. The midsegment is ½ the length of the third side. A R C S B RS 1 2 AB Midsegment Demonstration January 19, 2016 Geometry 5.4 Midsegment Theorem 10

11 Using the midsegment properties C DE is a midsegment of ABC D 12? E A 24 B January 19, 2016 Geometry 5.4 Midsegment Theorem 11

12 Using the midsegment properties C DE is a midsegment of ABC D 5 E A 10? B January 19, 2016 Geometry 5.4 Midsegment Theorem 12

13 Example 1: C D 6? E A F 12 B January 19, 2016 Geometry 5.4 Midsegment Theorem 13

14 Example 1: C? 8 D 6 E 4 A F 12 B January 19, 2016 Geometry 5.4 Midsegment Theorem 14

15 Example 1: C 8 D 6 E 10 5? 4 A F 12 B January 19, 2016 Geometry 5.4 Midsegment Theorem 15

16 Example 1: C 30 The perimeter of the outer triangle is. 8 D 6 E A F B The perimeter of the inner triangle is. January 19, 2016 Geometry 5.4 Midsegment Theorem 16

17 Example 1: C The perimeter of the outer triangle is. D E 4 A F 12 B January 19, 2016 Geometry 5.4 Midsegment Theorem 17

18 Example 2 Solve for x. DE is a midsegment of ABC C DE = ½ AB 3x 8 (10x 4) 1 2 3x 8 5x 2 D 3x E 6 2x x 3 A F 10x B January 19, 2016 Geometry 5.4 Midsegment Theorem 18

19 Your Turn DF is a midsegment of ABC Solve for x. C DF = ½ CB 4x 6 (20x 12) 1 2 4x 6 10x x x 2 A D 14 F 20x E B January 19, 2016 Geometry 5.4 Midsegment Theorem 19

20 Some Old, and Important, Formulas Midpoint x x y y, Distance 2 2 d x x y y Slope y y m x x January 19, 2016 Geometry 5.4 Midsegment Theorem 20

21 Coordinate Geometry On graph paper, draw RST R(0,0) S(2, 6) T(8, 0) Find M, the midpoint of RS. (1, 3) Find N, the midpoint of ST. (5, 3) S(2, 6) M(1, 3) N(5, 3) Draw midsegment MN. R(0, 0) T(8, 0) January 19, 2016 Geometry 5.4 Midsegment Theorem 21

22 Coordinate Geometry Verify that MN is parallel to RT. Slope of MN m S(2, 6) Slope of RT m M(1, 3) N(5, 3) R(0, 0) T(8, 0) Slopes are equal: Lines are parallel. January 19, 2016 Geometry 5.4 Midsegment Theorem 22

23 Coordinate Geometry Verify that MN = ½ RT. Length MN Length RT MN RT S(2, 6) M(1, 3) N(5, 3) R(0, 0) MN = ½ RT T(8, 0) January 19, 2016 Geometry 5.4 Midsegment Theorem 23

24 Coordinate Geometry The theorem is verified. S(2, 6) 4 M(1, 3) N(5, 3) R(0, 0) 8 T(8, 0) January 19, 2016 Geometry 5.4 Midsegment Theorem 24

25 A Proof In the kaleidoscope image, AE BE and AD CD. Show that CB DE. Because AE BE and AD CD, E is the midpoint of AB and D is the midpoint of AC by definition. Then DE is a midsegment of ABC by definition and CB DE by the Triangle Midsegment Theorem. January 19, 2016 Geometry 5.4 Midsegment Theorem 25

26 Summary A midsegment (midline) of a triangle is the segment between the midpoints of two sides. The midsegment is parallel to the third side. The midsegment is half the length of the third side. January 19, 2016 Geometry 5.4 Midsegment Theorem 26

27 Homework January 19, 2016 Geometry 5.4 Midsegment Theorem 27

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