Parallelogram, Point P divides AB in the ratio 2:1, lines AC and DP meet at Q and lines BQ and AD meet at R.?

Determine the images of P,Q and R under the Affine transformation t which maps A, D and C to (0,1), (0,0) and (1,0) respectively.

Parallelogram, Point P divides AB in the ratio 2:1, lines AC and DP meet at Q and lines BQ and AD meet at R.?
get some graph paper





mark the points A, C, and D using a large scale...


aka use 5 blocks = 1





then finish the parallelogram by marking B in the only logical place: (1,1) %26lt;-- this will be obvious when you see A, C, and D





now, what you just plotted are the images of ACD after the flip, so what you're going to find for P Q and R will also be the images... so what we find here will be your answers








next plot point P. i plotted it at (2/3 , 1) since i read it is as from A to P = twice from P to B.


everything from here forward is contingent upon this.








next, draw AC and DP and then find their equations


AC should be pretty simple: y-intercept of 1, slope of -1


y = -x+1 or y = 1-x





DP, find your slope, which will be 3/2 and since it crosses at the origin, your equation is y = 3/2 x





now, the point of intersection, point Q is the point at which the equations AC and DP are equal, so set them equal to each other and solve for x


1-x = 3/2 x


1 = 3/2 x + x


1 = 5/2 x


x = 2/5


plug that in for x in either equation and find y


y = 1-x = 1- 2/5 = 3/5





point Q: (2/5 , 3/5)











next, find the equations of the lines BQ and AD


you'll find them to be: BQ: y=(2/3)x + (1/3)





and AD is the y-axis, or x = 0


they intersect at point R


2 ways to do this: solve BQ for x then set equal to AD


or, since AD: x=0, gives you the value of x at the intersection, substitute that in for x in BQ and solve for y.





y = (2/3)(0) + (1/3)


y = 1/3





point R = (0 , 1/3)








P: (2/3,1)


Q: (2/5,3/5)


R: (0,1/3)


_____





note: i assume you know how to find the equation of a line given its graph... so i left that for you to do... :)
Reply:R divides AD in the ratio 2:1
Reply:You did not mentioned D point.