http://www.ck12.org Chapter 5. Relationships with Triangles
- FindNMandQO.
- If the midpoints of the sides of a triangle areA( 1 , 5 ),B( 4 ,− 2 ), andC(− 5 , 1 ), find the vertices of the triangle.
Answers:
- Use the midpoint formula 3 times to find all the midpoints.
LandM=
(
4 +(− 2 )
2 ,5 +(− 7 )
2)
= ( 1 ,− 1 ), pointOLandN=
(
4 +(− 8 )
2 ,5 + 3
2)
= (− 2 , 4 ), pointQMandN=
(
− 2 +(− 8 )
2 ,− 7 + 3
2)
= (− 5 ,− 2 ), pointPThe graph would look like the graph below.
- 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.
- 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
Note thatQOishalfofNM.
- The easiest way to solve this problem is to graph the midpoints and then apply what we know from the Midpoint
Theorem.