Geometry question?

Question

I cannot figure out how to display this numerically, someone pls help.

A (1.3) B(5,7) C(4,8) and D (a,b) form a rectangle find a,b

Now I have come up with so far the rules, that if it is a rectangle then AB and CD must be parallel so gradient AB = gradient CD which has left me with

(7-3) 8-b
------ = ------
5-1 4-a

4/4 = 8-b/4-a

Now I know here that obviously b is 4 and a is 0, but from the above a could equal 2 and b could equal 6 for all the layma knows

So then i went and said, for it to be a rectangle AB must be perpendicular to BC, I found ab gradient to be 1, so therefore BC gradient must be -1 and as a result

So therefore I said that 7-3/5-1 * b-3/a-1 must equal -1

therefore 1* b-3/a-1 must equal -1.

Now the only number that multiplies 1 to acheive -1 is -1

so b-3/a-1 = -1

but i have no idea where to take it from here, do I form a simualtaneous equation to work the rest out or what??

Answer 1

I would do it like this
The slope of AB is 1

The slope of BC is -1 so BC is perpendicular AB

The missing side must be perpendicular to AB through A
y-3 =-(x-1) This is AD

CD is the paralel to AB through C

y-8 = x-4 This is CD

Solving the system
y=-x+4
y=x+4 -x+4 = x+4 so x=0 and y=4 (0,4) is point D

Answer 2

A (1.3) B(5,7) C(4,8) and D (a,b) form a rectangle find a,b
If it is a rectangle, the slope of AB must be the negative reciprocal of the slope of DA.
m(AB) = (7 - 3)/(5 - 1) = 4/4 = 1, so
m(DA) = -1, and
y - 8 = x - 4
y - 3 = - x + 1
2y - 11 = - 3
2y = 8
y = 4 (b)
x = 4 - 8 + 4 = 0
(a,b) = (0,4)

Answer 3

well simply u found ur first equation:
(7-3)/(5-1)=(8-b)/(4-a) which is equavalent with this equation: a-b=-4 (1)
and the other equation u found is (b-3)/(a-1)=-1
which is equavalent with this equation:
a+b=4 (2)
by adding (1) and (2) we have
(1)+(2)=>a+b+a-b=0 => 2a=0 so a=0 and by putting in(1) or (2) we have b=4 it is ur answer
D(0,4) .

Answer 4

I tried to graph this. I just figure that point C is left 1 and up 1 from point C. So, from point A can't you just go left 1 and up 1 also? So, point D is (0,4) ?