#### Transcript Midsegment Theorem and Coordinate Proofs

```Ch 5.1
We will look at the properties of a
midsegment and use them to
solve for desired distances
Learn a new kind of proof
A midsegment connects 2 midpoints of a
triangle together.
There are 3 midsegments for each triangle.
Each midsegment is parallel to one side of a
triangle.
The midsegment is half the length of the
side of the triangle that it is parallel to.
AB
LM
AC
½ AC
MB = LN
2(10x – 9) = 8x + 12
20x – 18 = 8x + 12
12x = 30
X = 6, DE = 8(6) + 12 = 60
2(7x - 6) = 9x + 8
14x – 12 = 9x + 8
5x = 20
X = 4, EK = ½ (9(4) + 8) = 22
2(3x + 11) = 18x - 6
6x + 22 = 18x - 6
28 = 12x
X = 2.33, JK = 3(2.33) + 11= 17.99
 A coordinate proof uses the coordinate plane to make
generalizations.
 We place a shape onto the coordinate system with
dimensions that have variables as the coordinates.
 Then whatever is concluded works as a proof since it
is generalized for any value.
Place a square on the coordinate plane find
the length of its diagonal and then its
midpoint. What can you conclude?
dis tance  (b  0) 2  (a  0) 2  b 2  a 2
a0 b0
a b
m idpoint  (
,
)( , )
2
2
2 2
dis tance  (2q  0) 2  (2r  0) 2  4q 2  4r 2  4(q 2  r 2 )  2 q 2  r 2
dis tance  (q  p  p) 2  (r  0) 2  q 2  r 2
```