A(1,3),B(5,5),C(9,?),D(?,-1)

Question

Find the ordered pairs

Answer 1

We don't know what shape figure ABCD is.

Let's assume it's a parallelogram. Slope of AB=(5-3)/(5-1)=2/4=½.

Let C be (9,a) and D be (b,-1). Slope of CD=(-1-a)/(b-9) which must also be ½, because AB and CD are parallel.

So (-1-a)/(b-9)=1/2, -2-2a=b-9, 2a+b=7, so b=7-2a.

BC and AD are also parallel. Their slopes must therefore be equal so:

(a-5)/(9-5)=(-1-3)/(b-1), 

(a-5)/4=-4/(b-1), (a-5)(b-1)=-16, so substituting b=7-2a:

(a-5)(7-2a-1)=-16, (a-5)(6-2a)=6a-2a²-30+10a=-2a²+16a-30=-16,

2a²-16a+14=0, a²-8a+7=0=(a-1)(a-7), so a=1 or 7 and b=7-2a=5 or -7.

Therefore C is (9,1) and D is (5,-1); or C is (9,7) and D is (-7,-1).

We can reject C(9,7) and D(-7,-1), because it would make A, B, C and D collinear (on the same straight line).

Therefore the solution must be C(9,1) and D(5,-1). ABCD is a parallelogram.