Question 1197815
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Let
A = ( 4, 5 )
B = ( -7, -6 )
C = ( 10, -3 )
represent the vertices of the triangle.


Furthermore we'll have
P = midpoint of AB
Q = midpoint of BC
R = midpoint of AC


Drawing:
{{{drawing(500,500,-12,12,-10,10,
graph(500,500,-12,12,-10,10,0),
line(4,5,-7,-6),
line(4,5,10,-3),
line(-7,-6,10,-3),
circle(4,5,0),
circle(4,5,0.1),
circle(4,5,0.2),
circle(-7,-6,0),
circle(-7,-6,0.1),
circle(-7,-6,0.2),
circle(10,-3,0),
circle(10,-3,0.1),
circle(10,-3,0.2),
circle(-1.5,-0.5,0),
circle(-1.5,-0.5,0.1),
circle(-1.5,-0.5,0.2),
circle(1.5,-4.5,0),
circle(1.5,-4.5,0.1),
circle(1.5,-4.5,0.2),
circle(7,1,0),
circle(7,1,0.1),
circle(7,1,0.2),
locate(4,5+0.8,"A"),
locate(-7,-6-0.25,"B"),
locate(10,-3-0.25,"C"),
locate(-1.5,-0.5-0.25,"P"),
locate(1.5,-4.5-0.25,"Q"),
locate(7,1+0.8,"R")
)}}}


To compute the midpoint, we add up the x coordinates and divide in half. 
Let's do this for points A(4,5) and B(-7,-6)
Add: 4 + (-7) = -3
Cut in half: -3/2 = -1.5
This is the x coordinate of point P


Do the same for the y coordinates
Add: 5 + (-6) = -1
Cut in half: -1/2 = -0.5


Point P is located at (-1.5, -0.5)


Here's what you should get for points P through R
P = (-1.5, -0.5)
Q = (1.5, -4.5)
R = (7, 1)
I'll skip the steps showing how to locate Q and R, but you'd follow similar steps when I found P.


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Definition: A midsegment is a segment connecting two midpoints of a triangle.


We have these three midsegments
PQ
PR
QR


Let's use the distance formula to find the distance from B(-7,-6) to C(10,-3)


(x1,y1) = (-7,-6) and (x2,y2) = (10,-3)


{{{d = sqrt( (x1-x2)^2 + (y1-y2)^2 )}}}


{{{d = sqrt( (-7-10)^2 + (-6-(-3))^2 )}}}


{{{d = sqrt( 298 )}}}


Segment BC is exactly {{{sqrt(298)}}} units long.


Now let's find the distance from P(-1.5, -0.5) to R(7, 1)
(x1,y1) = (-1.5,-0.5) and (x2,y2) = (7,1)


{{{d = sqrt( (x1-x2)^2 + (y1-y2)^2 )}}}


{{{d = sqrt( (-1.5-7)^2 + (-0.5-1)^2 )}}}


{{{d = sqrt( 74.5 )}}}


Midsegment PR is exactly {{{sqrt(74.5)}}} units long.


Compute the ratio of BC and PR.


{{{(BC)/(PR) = sqrt( 298 )/sqrt( 74.5 )}}}


{{{(BC)/(PR) = sqrt( 298/74.5 )}}}


{{{(BC)/(PR) = sqrt( 4 )}}}


{{{(BC)/(PR) = 2}}}


We can then rearrange that like so
{{{(BC)/(PR) = 2}}}


{{{BC = 2*PR}}}


{{{PR = (1/2)*BC}}}
Showing us that midsegment PR is exactly half as long compared to BC.


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Repeat these sets of steps to show these two other facts
{{{PQ = (1/2)*AC}}}


{{{QR = (1/2)*AB}}}
I'll let the student do these.
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