# Triangle geometry find a side length

1. Jun 10, 2012

### Wildcat

1. The problem statement, all variables and given/known data
In triangle ABC, the median from C meets AB at D. Through M, the midpoint of CD, line AM is drawn meeting CB at P. If CP=4, find CB.

2. Relevant equations

3. The attempt at a solution
I constructed this drawing on GSP and found CB to be 12. I'm trying to show similarity between some triangles in the drawing but can't find any. I would like to know how to solve this without GSP. any ideas??

2. Jun 10, 2012

### tiny-tim

Hi Wildcat!

Hint: areas.

3. Jun 10, 2012

### Wildcat

Ok, I don't see where I can calculate any areas with the information I have unless I'm missing something. Will I need to construct another segment?

4. Jun 11, 2012

### tiny-tim

Hi Wildcat!

(just got up :zzz:)
Yes.

Divide the triangle into triangles, call two of the unequal areas "p" and "q", and add them all up.

5. Jun 11, 2012

### VietDao29

Hi Wildcat,

Apart from the area method, there's still another way to tackle this problem. It's to use mid-segment of a triangle (it's the line segment that connects the two midpoints of any 2 sides of a triangle).

There are 2 theorems about mid-segment you should remember is:
Given $\Delta ABC$
• If M, and N are respectively the midpoints of AB, and AC then $MN = \frac{1}{2}BC$, and $MN // BC$.
This theorem means that the mid-segment of a triangle is parallel to the opposite side, and is half of it.​
• If a line passes through the midpoint of one side, and is parallel to the second side, then it also passes through the midpoint of the other side.

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Hint: Look closely at the 2 triangles $\Delta ABP$, and $\Delta CDQ$