In rectangle ABCD, we have A=(6,-22), B=(2006,178) , and D=(8,y), for some integer y. What is the area of rectangle ABCD?

The number of distinct points in the xy-plane common to the graphs of (x + y - 5)(2x - 3y + 5) = 0 and (x -y + 1)(3x + 2y - 12) = 0 is

SaltyGrandma Jan 26, 2019

#1**+3 **

In rectangle ABCD, we have A=(6,-22), B=(2006,178) , and D=(8,y), for some integer y. What is the area of rectangle ABCD?

The slope from A to B is [ 178 - -22] / [ 2006 - 6 ] = 200/ 2000 = 1/10

So...the slope between A and D will have a negative reciprocal slope....so we have

[ y - - 22 ] / [ 8 - 6 ] = -10

[ y + 22] [ 2] = -10

y + 22 = -20

y = -2

So D = (8, - 2)

And the distance from A to B = sqrt [ (2006 - 6)^2 + (-22 - 178)^2 ] = sqrt [ 2000^2 + 200^2 ] = sqrt (4040000)

And the distance from A to D is sqrt [ (8 - 6)^2 + (-22 - - 2)^2 ] = sqrt [ (4 + 400] = sqrt (404)

So....the area of ABCD = sqrt (4040000)*sqrt (404) = sqrt (404000 * 404 ) = 40,400 units^2

CPhill Jan 26, 2019

#2**+3 **

The number of distinct points in the xy-plane common to the graphs of (x + y - 5)(2x - 3y + 5) = 0 and (x -y + 1)(3x + 2y - 12) = 0

Notice that this will be true whenever

x + y - 5 = 0 and

x - y + 1 = 0 add these

2x - 4 = 0

x = 2

And y = 3

And note that these two values make (2x - 3y + 5) = 0 and ( 3x + 2y - 12) = 0

So.....the common point to both graphs is (2,3)

See the graphs here : https://www.desmos.com/calculator/rwobs7w4hu

CPhill Jan 26, 2019