Question

Points A(-2,4),B(1,3),C(4,1) form a parallelogram what are the coordinates of D

Answer 1

The parametric equation of the line from A, parallel to BC
is
h(s)= {-2 + 3 s, 4 - 2 s}

Answer 2

The parametric equation of the line from D, parallel to BA
is
g(s)={4 - 3 s, 1 + s}

Answer 3

These two line intersect when s=1, hence D= h(1)=(1,2)

Answer 4

at least you should put A, B, and C on graph paper, and decide where to put D. Then you can use simple coordinate geometry of lines and the distance formula, maybe, to find the coordinates needed for D. Since you want a parallelogram, you know that there are two pair of parallel lines, you know two points define any line...


The question seems slightly open-ended. Arbitrarily picking BC as one of the sides, and AB as the other side, the two sides meeting at point B, the slope for the side BC is %28-1-3%29%2F%284-1%29=-4%2F3.

The parallelogram needs another side, AD, which has the same slope, -4/3, as line BC. Note too that line AD obviously contains point A. We have the coordinates for point A, being (-2,4). We can use the POINT-SLOPE form for a linear equation to get line AD.

Line AD is y-v=m%28x-u%29, where m is slope, (u,v) is a point known on the line.
y-4=%28-4%2F3%29%28x-%28-2%29%29
y-4=-%284%2F3%29x-8%2F3
y=-%284%2F3%29x%2B4-8%2F3-------(MISTAKE STEP HERE. SHOULD HAVE BEEN highlight%28y=-%284%2F3%29x%2B12%2F3-8%2F3%29, and then highlight%28y=-%284%2F3%29x%2B4%2F3%29)
y=-%284%2F3%29x-4%2F3, line AD.

You also want to know line DC, because this will intersect line AD, at point D. Line DC will need to be parallel to line AB, so we must find slope for AB, and use known point C in POINT-SLOPE form for a line to find line DC.

Slope AB is slope DC, %283-4%29%2F%281-%28-2%29%29=-1%2F3, this slope.
y-v=m%28x-u%29 which is generalized point-slope form; see above.
y-%28-1%29=-%281%2F3%29%28x-4%29
y=-%281%2F3%29x%2B4%2F3+-1
y=-%281%2F3%29x%2B1%2F3, line DC.

FINALLY, at what point D, do lines AD and DC intersect?
Equating their formulas for y,