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Sec 6.1 Median

Sec 6.1 Median. Objective. Identify and construct medians in triangles. Medians. Picture:. Both sides are congruent. Median. vertex to midpoint. How many medians can a triangle have?. 3 medians. Median. vertex to midpoint. Midpoint-.

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Sec 6.1 Median

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  1. Sec 6.1 Median

  2. Objective • Identify and construct medians in triangles

  3. Medians Picture: Both sides are congruent Median vertex to midpoint

  4. How many medians can a triangle have? 3 medians Median vertex to midpoint

  5. Midpoint- • If you have a midpoint- then the segments on both sides are CONGRUENT! • That is why you will see the “tick marks” • For the measure of the entire line- add both sides! EQ: What are the differences between medians, altitudes, and perpendicular bisectors?

  6. M D P C 9 N 1. What is NC if NP = 18? 2. If DP = 7.5, find MP. 15

  7. You Try the Following: A B C D 14 E 1.What is ED if DC = 14? 2.What Is AC if BC is 9? 18 3.If BC = 3, find AC. 6

  8. So if you are given the length of the entire side, how do you find a missing segment? • If you are given the length of a segment, how to you find the entire side? • If you have a median- what do you know about each side?

  9. A E B D C If CD = 2x + 5, BD = 4x – 1, and AE = 5x –2, find BE. BD = CD AE = BE BE = 13 4x – 1= 2x + 5 BE = 5x – 2 BE = 5(3) – 2 2x = 6 x = 3

  10. The intersection of the medians is called the CENTROID. Draw the Picture: How many medians does a triangle have?

  11. Refer to the figure on the right. Imagine you have a triangular metal plate, and try and balance it on a point - say a pencil tip. -Once you have found the point at which it will balance, that is the centroid. Also called, 'center of gravity’ or 'center of mass' EQ: What are the differences between medians, altitudes, and perpendicular bisectors?

  12. Concurrent: • When three or more lines or segments meet at the same point.

  13. Theorem 6-1 Distance from vertex to centroidis twicethe distance fromcentroid to midpoint. Draw Picture 2x x

  14. Vertex to Centroid  LONGER (2x) Centroid to Midpoint  shorter (x) x + 2x = 3x = whole median EQ: What are the differences between medians, altitudes, and perpendicular bisectors?

  15. C How much is CX? D CX = 2(XF) E X CX = 2(13) 13 B A F CX = 26

  16. Quick Assessment • What is a median? • What is a centroid? • What does concurrent mean? • What is the vertex? • What is a midpoint? EQ: What are the differences between medians, altitudes, and perpendicular bisectors?

  17. Centroid Lab • Lab- With partners • Worksheet

  18. C How much is XD? D AX = 2(XD) E X 18 18 = 2(XD) B A F 9 = XD

  19. Ex: 1 In ABC, AN, BP, and CM are medians. If EM = 3, find EC. C N EC = 2(3) P E EC = 6 B M A EQ: What are the differences between medians, altitudes, and perpendicular bisectors?

  20. Ex: 2 In ABC, AN, BP, and CM are medians. C If EN = 12, find AN. AE = 2(12)=24 N P E AN = AE + EN B AN = 24 + 12 M A AN = 36 EQ: What are the differences between medians, altitudes, and perpendicular bisectors?

  21. Warm-UP • With an index card write one positive thing about me/my teaching/ etc.

  22. Sec 6.2 Altitudes and Perpendicular Bisectors

  23. Objectives • Identify and construct altitudes and perpendicular bisectors in triangles

  24. Altitude Picture: Altitude vertex to opposite side and perpendicular

  25. Altitude The altitude is the “true height” of the triangle. EQ: What are the differences between medians, altitudes, and perpendicular bisectors?

  26. Tell whether each red segment is an altitude of the triangle. The altitude is the “true height” of the triangle. YES NO YES

  27. Tell whether each red segment is an altitude of the triangle. The altitude is the “true height” of the triangle. Draw examples page 235

  28. Perpendicular Bisector Both sides are congruent- make sure you see this or it is NOT a perpendicular bisector Picture: Perpendicular Bisector midpoint and perpendicular

  29. Tell whether each red segment is an perpendicular bisector of the triangle. NO NO YES

  30. Can you have both? • Can it be both an altitude and perpendicular bisector? • Help Me Draw Examples:

  31. Quick Assessment • What is the difference between altitude and perpendicular bisector? • What is an altitude? • What is a perpendicular bisector? • How can the segment be both- altitude and perpendicular bisector? EQ: What are the differences between medians, altitudes, and perpendicular bisectors?

  32. Graphic Organizer • Compare and Contrast: Perpendicular Bisector Altitude Median Angle Bisector

  33. Medians Picture: Both sides are congruent Median vertex to midpoint

  34. Theorem 6-1 Distance from vertex to centroidis twicethe distance fromcentroid to midpoint. Draw Picture 2x x

  35. Vertex to Centroid  LONGER (2x) Centroid to Midpoint  shorter (x) x + 2x = 3x = whole median EQ: What are the differences between medians, altitudes, and perpendicular bisectors?

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