## Day 2 Medians of triangles

### Objectives:

1) Definition of a median and why important

2) Proper construction of a median

3) Use the median of triangle to solve algebra problems regarding
the triangle

4) Definition of centroid

5) Relationship of the length from vertex to centroid and centroid
to midpoint of opposite side

### 1) Definition of median

### A segment that joins a vertex of the
triangle and the midpoint of the side opposite the vertex

### Can be used to find the distance between two places.

### Provide examples of medians

### 2) Construct the medians

### First show how to construct with compass and straight edge
(reminding the students of the lesson on bisecting a segment)

### Then show how to use GSP to construct the median.

### Construct the median together with the students, then have
the students construct the remaining medians...have them figure
out what determines the number of medians a triangle has.

### 3) Using their constructions, provide them problems with finding
the lengths of different segments given the triangles and medians.
For example: In triangle ABC, segment BD is a median, find the
length of segment AD if we know that AC = 11

### Practice Problem: In triangle MNP, segments
MC and ND are medians

a) What is the length of NC if NP = 9

### b) If DP = 7.5, find the length of MP

###

### Using the same concept link these ideas to algebra

### Practice Problem: In triangle RST, segments
RP and SQ are medians. If RQ = 7x 1, SP = 5x 4, and
QT = 6x + 9, find the length of PT

### 4) Definition of Centroid

The medians of triangle ABC intersect at
a common point called the centroid.

When three or more lines or segments meet
at the same point we say they are concurrent.

Have the students construct various triangles and their medians
to explore whether the medians of all triangles are congruenta
perfect way to do this is with GSP.

### 5) Have the students explore the relationship between the
length of the segment from the vertex to the centroid and the
length of the segment from the centroid to the midpoint, using
either GSP or a straightedge.

Then cover the following theorem:

The length of the segment from the vertex to the centroid is twice
the length of the segment from the centroid to the midpoint.