http://www.ck12.org Chapter 5. Relationships with Triangles
CK-12 Foundation: Chapter5MidsegmentTheoremB
Concept Problem Revisited
To the left is a picture of the 4thfigure in the fractal pattern. The number of triangles in each figure is 1, 4, 13, and
- The pattern is that each term increase by the next power of 3.
Vocabulary
A line segment that connects two midpoints of the sides of a triangle is called amidsegment. Amidpointis a point
that divides a segment into two equal pieces. Two lines areparallelif they never intersect. Parallel lines have slopes
that are equal. In a triangle, midsegments are always parallel to one side of the triangle.
Guided Practice
The vertices of 4 LMNareL( 4 , 5 ),M(− 2 ,− 7 )andN(− 8 , 3 ).
- Find the midpoints of all three sides, label themO,PandQ. Then, graph the triangle, it’s midpoints and draw in
the midsegments. - Find the slopes ofNMandQO.
- 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.