1. ## vertices??? and transformation...

the points A(-5,1), B(-1,-1) and C(2.5) are three vertices of a rectangle ABCD

1. what are the coordinates of D.
2.length of the segment AB
3.area of the rectangle ABCD
4.find the equation of the line in which AD is the reflection of BC
5.describe a transformation which maps A onto D and B onto C.

2. to find D..it's a rectangle..so AD=BC

X+5=3....X=-2
Y-1=6......Y=7...........D(-2,7)

3. AB=square root((-1+5)squared+(-1-1)squared)
=square root(16+4)
=square root(20)
=2 square root 5

4. 1. what are the coordinates of D.
Basing on the line BC, where, from B, point C went 3 units to the right and 6 units up,
D will do the same basing from A, so,
D = ((-5 +3),(1 +6)) = (-2,7) --------------answer.

2.length of the segment AB
AB = sqrt[(-1 -(-5))^2 +(-1 -1)^2] = sqrt[16 +4] = sqrt(20) = 2sqrt(5) -------answer.

3.area of the rectangle ABCD
area, K, of ABCD = (AB)(BC)
K = sqrt(20)*sqrt[(2 -(-1))^2 +(5 -(-1))^2]
K = sqrt(20)*sqrt(45)
K = sqrt(900)

4.find the equation of the line in which AD is the reflection of BC
The axis of reflection must be the line connecting the midpoints of AB and CD.
Midpoint of AB = ((-5 +(-1))/2),((1 +(-1))/2) = (-3,0)
Midpoint of CD =((-2 +2)/2),(7 +5)/2) = (0,6)
So, using point-slope form of the equation of the line,
m = (6-0)/(0 -(-3)) = 2
Using point (-3,0),
(y -0) = 2(x -(-3))
y = 2x +6 ------------------axis of reflection, answer.

5.describe a transformation which maps A onto D and B onto C.
The whole thing moves 3 units to the right and 6 units up.
Or, all of the new coordinates are now in the form ((x+3),(y+6)).