Triangle geometry find a side length

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Homework Help Overview

The problem involves triangle geometry, specifically focusing on the relationships between sides and medians within triangle ABC. The original poster is tasked with finding the length of side CB given certain conditions involving points D, M, and P.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • The original poster attempts to use geometric software to visualize the problem and has found a potential solution. They express interest in alternative methods, particularly in showing similarity between triangles. Some participants suggest using area calculations and constructing additional segments to aid in the solution.

Discussion Status

The discussion is ongoing, with participants exploring various methods to approach the problem. Some guidance has been offered regarding the use of area and mid-segment theorems, but there is no explicit consensus on a single method or solution yet.

Contextual Notes

Participants are working with the constraints of the problem as stated, and there is a mention of needing to construct additional segments to facilitate calculations. The original poster is also seeking to avoid reliance on geometric software.

Wildcat
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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??
 
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Hi Wildcat! :smile:

Hint: areas. :wink:
 
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?
 
Hi Wildcat! :smile:

(just got up :zzz:)
Wildcat said:
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. :smile:
 
Wildcat said:

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,
 

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