Integrated Math 2 / Unit 6: Similarity
6.4 Triangle Proportionality and Midsegments
Notes
A line parallel to one side of a triangle creates a smaller similar triangle.
- Triangle Proportionality Theorem: if $DE \parallel BC$ with $D$ on $AB$ and $E$ on $AC$, then $\displaystyle \frac{AD}{DB} = \frac{AE}{EC}$.
- The converse is also true: if those ratios match, the segment is parallel to the side.
- A midsegment joins two midpoints. It is parallel to the third side and half as long.
- Example: $AD = 4$, $DB = 6$, $AE = 6$: $\displaystyle \frac{4}{6} = \frac{6}{EC}$, so $EC = 9$.
Practice
In triangle $ABC$, $DE \parallel BC$ with $D$ on $AB$ and $E$ on $AC$. If $AD = 4$, $DB = 6$ and $AE = 6$, find $EC$.
$\displaystyle \frac{AD}{DB} = \frac{AE}{EC}$ gives $\displaystyle \frac{4}{6} = \frac{6}{EC}$, so $EC = 9$.
A midsegment of a triangle is parallel to the side of length 18. How long is the midsegment?
A midsegment is half the length of the side it is parallel to.
In triangle $ABC$, $D$ is on $AB$ and $E$ is on $AC$ with $AD = 3$, $DB = 5$, $AE = 6$, $EC = 10$. Is $DE \parallel BC$?
The converse of the proportionality theorem says matching ratios make the segment parallel.
Ready for more? Take the Unit 6 test.