Midpoints of Triangle Sides

Points X(-1,1),Y(3,5), and Z(17,1) are the midpoints of the sides of triangle ABC. Find the coordinates of the points A,B, and C. (Labels A,B, and C are arbitrarily assigned to coordinates).
Source: NCTM Mathematics Teacher 2008

SOLUTION

Draw a first line passing through Y and parallel to \overleftrightarrow{XZ}. Equation of the first line
y=5\qquad\qquad (1)
Slope of \overleftrightarrow{XY}
\dfrac{1-5}{-1-3}=1
Draw a second line passing through point Z and parallel to \overleftrightarrow{XY}. Equation of the second line
y=1x+b
Substitute the values of Z(17,1) into the equation
1=17+b
b=-16
Equation of the second line
y=x-16\qquad (2)
The two lines intersect at point A(x,y) the coordinates of which satisfy both Eq. (1) and (2).
Solving for x yields
5=x-16
x=21
Coordinates of vertex A=(21,5).
Suppose Z(17,1) is the midpoint of side \overline{AB}. The midpoint formula yields the coordinates of vertex B(x,y)
\dfrac{x+21}{2}=17
x=13
\dfrac{y+5}{2}=1
y=-3
Coordinates of vertex B=(13,-3).
Similarly X(-1,1) is the midpoint of side \overline{BC}. In this case the midpoint formula yields the coordinates of B(x,y)
\dfrac{x+13}{2}=-1
x=-15
\dfrac{y-3}{2}=1
y=5
Coordinates of vertex C=(-15,5).

Answer: (21,5),(13,-3),(-15,5)

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About mvtrinh

Retired high school math teacher.
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