SOLUTION: The midpoints of the sides of a triangle are at (0,-3), (-1/2,7/2) and (-7/2,1/2). Find the coordinates of the vertices

Algebra ->  Coordinate-system -> SOLUTION: The midpoints of the sides of a triangle are at (0,-3), (-1/2,7/2) and (-7/2,1/2). Find the coordinates of the vertices      Log On


   

Question 780900: The midpoints of the sides of a triangle are at (0,-3), (-1/2,7/2) and (-7/2,1/2). Find the coordinates of the vertices
Answer by Edwin McCravy(20059) About Me  (Show Source):
You can put this solution on YOUR website!
In a message dated 9/5/2013 9:00:27 A.M. Eastern Daylight Time, AnlytcPhil@aol.com writes:
 (0,-3), (-1/2,7/2) and (-7/2,1/2)

Plot the points, and connect them:

 

These green lines are the mid-segments of the triangle we are trying
to determine the coordinates of.

We remember that a mid-segment of a triangle (which joins the 
midpoints of two sides of a triangle) is parallel to the third 
side of the triangle.  The mid-segment is also 1/2 of the third
side , but we do not need that fact.

Through each of those points we will find the equation of the line 
paralell to the mid-segment joining the other two points:

We find the slope of the mid-segment joining

(0,-3) and  (-1%2F2,7%2F2)

m = %28y%5B2%5D-y%5B1%5D%29%2F%28x%5B2%5D-x%5B1%5D%29

where (x1,y1) = (0,-3)
and where (x2,y2) = (-7%2F2,1%2F2)

m = %28%287%2F2%29-%28-3%29%29%2F%28%28-1%2F2%29-%280%29%29 = %287%2F2%2B3%29%2F%28-1%2F2-0%29 = %2813%2F2%29%2F%28-1%2F2%29 = 13%2F2÷-1%2F2 = 13%2F2×-2%2F1 = 13%2Fcross%282%29×-cross%282%29%2F1 = -13

Now we find the equation of the line through the third point (-7%2F2,1%2F2):

Point-slope formula:

y - y1 = m(x - x1)

where (x1,y1) = the third point (-7%2F2,1%2F2)

y - 1%2F2 = -13(x - (-7%2F2))

y - 1%2F2 = -13(x + 7%2F2)

y - 1%2F2 = -13x - 91%2F2)

    y = -13x - 90%2F2

    y = -13x - 45  

That line is the red one below:



-------------------

We find the slope of the mid-segment joining

(0,-3) and (-7%2F2,1%2F2)

m = %28y%5B2%5D-y%5B1%5D%29%2F%28x%5B2%5D-x%5B1%5D%29

where (x1,y1) = (0,-3)
and where (x2,y2) = (-7%2F2,1%2F2)

m = %28%281%2F2%29-%28-3%29%29%2F%28%28-7%2F2%29-%280%29%29 = %281%2F2%2B3%29%2F%28-7%2F2-0%29 = %281%2F2%2B6%2F2%29%2F%28-7%2F2%29 = %287%2F2%29%2F%28-7%2F2%29 = 7%2F2÷-7%2F2 = -1

Now we find the equation of the line through the third point (-1%2F2,7%2F2):

Point-slope formula:

y - y1 = m(x - x1)

where (x1,y1) = the third point (-1%2F2,7%2F2)

y - 7%2F2 = -1(x - (-1%2F2))

y - 7%2F2 = -1(x + 1%2F2)

y - 7%2F2 = -x - 1%2F2)

    y = -x + 6%2F2

    y = -x + 3  

 

That line is the second red one below:



-------------------

We find the slope of the mid-segment joining

(-1%2F2,7%2F2) and (-7%2F2,1%2F2)

m = %28y%5B2%5D-y%5B1%5D%29%2F%28x%5B2%5D-x%5B1%5D%29

where (x1,y1) = (0,-3)
and where (x2,y2) = (-7%2F2,1%2F2)

m = %28%281%2F2%29-%287%2F2%29%29%2F%28%28-7%2F2%29-%28-1%2F2%29%29 = %281%2F2-7%2F2%29%2F%28-7%2F2%2B1%2F2%29 = %28-6%2F2%29%2F%28-6%2F2%29 = %28-3%29%2F%28-3%29 = 1

Now we find the equation of the line through the third point  (0,-3), :

Point-slope formula:

y - y1 = m(x - x1)

where (x1,y1) = the third point (-1%2F2,7%2F2)

y - (-3)} = 1(x - 0)

y + 3 = x 

    y = x - 3  

 

That line is the second red one below:



Now we find the three points of intersection of the three pairs of
the lines

y = -13x - 45,  y = -x + 3, y = x - 3

Solve these three systems:

y = -13x - 45      y = -13x - 45      y = -x + 3   
y = -x + 3         y = x - 3          y = x - 3

You can do that.  They are

(-4,7),  (-3,-6), and (3,0)

Edwin