Side Splitter Theorem
A theorem to find sides of similar triangles

What is the side splitter theorem?

Answer:
The side splitter theorem states that if a line is parallel to a side of a triangle and intersect the other two sides, then this line divides those two sides proportionally.
The side splitter theorem is a natural extension of similarity ratio.
Example

The Side Splitter theorem states that
$
\frac{AC}{CE} = \frac{AB}{BD}
$


Is the proportion below true?


No, this example is not accurate. PM is obviously not parallel to OM.
Therefore, the side splitter theorem does not hold and is not true.
$ \frac{LP}{PO} \color{Red}{\ne} \frac{LM}{MN} $
$ \frac{LP}{PO} = \frac{LM}{MN} $


Is the proportion below true?


No, remember this theorem only applies to the segments that are 'split' or intercetped by the parallel lines.
$ \frac{VW}{WY} \color{Red}{\ne} \frac{WX}{YZ} $
Instead, you could set up the following proportion:
$ \frac{VW}{WY} = \frac{VX}{XZ} $
$ \frac{VW}{WY} = \frac{WX}{YZ} $


What if there are more than two parallel lines?

Answer: A corollary of the this theorem is that when three prallel lines intersect two transversals, then the segments intercepted on the transversal are proportional.
Example
Example 2
Practice Problems
Problem 1)
Find the length of VX by using the side splitter theorem.
To solve this problem, set up the following proportion and solve:
$
\frac{VW}{WY} = \frac{VX}{XZ}
\\
\frac{7}{14} = \frac{VX}{16}
\\
\frac{16 \cdot 7}{14} = VX
\\
VW = 8
$
Problem 2) Use the corollary to find the value of x in the problem pictured below.
Set up the proportion then solve for x:
Problem 3) Use the corollary to find the value of x in the problem pictured below.
Problem 4) Are the red segments pictured below parallel? (Picture not to scale)
If the red segments are parallel, then they 'split' or divide triangle's sides proportionally. However, when you try to set up the proption, you will se that it is not true:
$
\frac{10}{12} \color{Red}{\ne} \frac{8}{14}
$
Therfore, the red segments are not parallel
