Triangle geometry find a side length

[Wildcat]

Homework Statement

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.

Homework Equations

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??

 

[tiny-tim]

Hi Wildcat!

Hint: areas. ​

 

[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?

 

[tiny-tim]

Hi Wildcat!

(just got up :zzz:)

Will I need to construct another segment?

Yes.

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

 

[VietDao29]

Homework Statement

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.

Homework Equations

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??

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.

-------------------------------

So back to your problem,

Let d be a line that passes through D, and parallel to AM, it intersects BC at Q. Now, look at the 2 theorems above, what conclusion can you draw about P, and Q?

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

Cheers,