Sec 6.1 Median.

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Presentation transcript:

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? vertex to midpoint

4/28/2017 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?

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

14 18 6 A B C D E 1.What is ED if DC = 14? 2.What Is AC if BC is 9? 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

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?

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

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

Also called, 'center of gravity’ or 'center of mass' 4/28/2017 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?

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

Draw Picture Theorem 6-1 Distance from vertex to centroid is twice the distance from centroid to midpoint. 2x x

Vertex to Centroid  LONGER (2x) 4/28/2017 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?

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

Quick Assessment What is a median? What is a centroid? 4/28/2017 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?

Centroid Lab Lab- With partners Worksheet

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

In ABC, AN, BP, and CM are medians. 4/28/2017 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?

In ABC, AN, BP, and CM are medians. 4/28/2017 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?

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

Sec 6.2 Altitudes and Perpendicular Bisectors

Objectives Identify and construct altitudes and perpendicular bisectors in triangles

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

The altitude is the “true height” of the triangle. 4/28/2017 Altitude The altitude is the “true height” of the triangle. EQ: What are the differences between medians, altitudes, and perpendicular bisectors?

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

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

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

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

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

4/28/2017 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?

Graphic Organizer Compare and Contrast: Perpendicular Bisector Altitude Median Angle Bisector

Medians Picture: Both sides are congruent Median vertex to midpoint

Draw Picture Theorem 6-1 Distance from vertex to centroid is twice the distance from centroid to midpoint. 2x x

Vertex to Centroid  LONGER (2x) 4/28/2017 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?