http://www.ck12.org Chapter 5. Relationships with Triangles
Example 4:Find the slopes ofNMandQO.
Solution:The slope ofNMis−− 2 −^7 (−−^38 )=− 610 =−^53.
The slope ofQOis 1 −−^1 (−−^42 )=−^53.
From this we can conclude thatNM||QO. If we were to find the slopes of the other sides and midsegments, we
would findLM||QPandNL||PO.This is a property of all midsegments.
Example 5:FindNMandQO.
Solution:Now, we need to find the lengths ofNMandQO. Use the distance formula.
NM=
√
(− 7 − 3 )^2 +(− 2 −(− 8 ))^2 =
√
(− 10 )^2 + 62 =
√
100 + 36 =
√
136 ≈ 11. 66
QO=
√
( 1 −(− 2 ))^2 +(− 1 − 4 )^2 =
√
32 +(− 5 )^2 =
√
9 + 25 =
√
34 ≈ 5. 83
From this we can conclude thatQOishalfofNM. If we were to find the lengths of the other sides and midsegments,
we would find thatOPishalfofNLandQPishalfofLM.This is a property of all midsegments.
The Midsegment Theorem
The conclusions drawn in Examples 4 and 5 can be generalized into the Midsegment Theorem.
Midsegment Theorem:The midsegment of a triangle is half the length of the side it is parallel to.
Example 6:Mark everything you have learned from the Midsegment Theorem on 4 ABCabove.
Solution:Let’s draw two different triangles, one for the congruent sides, and one for the parallel lines.