Answer:
Explanation:
Step 1
1 of 5
Graph the three points and connect them to create the mid segments of the triangle:
Use midsegments of triangles in the coordinate plane.
Use the Triangle Midsegment Theorem to fi nd distances.
Using the Midsegment of a Triangle
A midsegment of a triangle is a segment that connects the
midpoints of two sides of the triangle. Every triangle has
three midsegments, which form the midsegment triangle.
The midsegments of △ABC at the right are MP —, MN —,
and NP —. The midsegment triangle is △MNP.
Using Midsegments in the Coordinate Plane
In △JKL, show that midsegment MN — is parallel to JL —
and that MN =
1
—2 JL.
SOLUTION
Step 1 Find the coordinates of M and N by fi nding
the midpoints of JK — and KL —.
M (—
−6 + (−2)
2
, —
1 + 5
2
) = M ( −8 —
2
, 6
—
2
) = M(−4, 3)
N (—
−2 + 2
2
, —
5 + (−1)
2
) = N ( 0
—
2
, 4
—
2
) = N(0, 2)
Step 2 Find and compare the slopes of MN — and JL —.
slope of MN —= —
2 − 3
0 − (−4) = −
1
—
4
slope of JL — = —
−1 − 1
2 − (−6) = −
2
—
8
= −
1
—
4
Because the slopes are the same, MN — is parallel to JL —.
Step 3 Find and compare the lengths of MN — and JL —.
MN = √——
[0 − (−4)]2 + (2 − 3)2 = √—
16 + 1 = √—
17
JL = √———
[2 − (−6)]2 + (−1 − 1)2 = √—
64 + 4 = √—
68 = 2 √—
17
JL = √———
[2 − (−6)]2 + (−1 − 1)2 = √—
64 + 4 = √—
68 = 2 √—
17
Because √—
17 =
1
—2
(2 √—
17 ) , MN =
1
—2 JL.
Monitoring Progress onitoring Progress Help in English and Spanish at BigIdeasMath.com
Use the graph of △ABC.
1. In △ABC, show that midsegment DE — is parallel
to AC — and that DE =
1
—2 AC.
2. Find the coordinates of the endpoints of
midsegment EF —, which is opposite AB —. Show
that EF — is parallel to AB — and that EF =
1
—2 AB.
READING
In the fi gure for Example 1,
midsegment MN — can be
called “the midsegment
opposite JL —.”
midsegment of a triangle,
p. 330
Previous
midpoint
parallel
slope
coordinate proof
Core Vocabulary Core Vocabulary
A C
P
N
M
B
x
y
4
6
−2
−6 −4 −2
M
N
K(−2, 5)
J(−6, 1)
L(2, −1)
x
y
−4
−6
−2 2 4 6
D
E
C(5, 0)
B(−1, 4)
A(