Sorry, i forgot to add a picture in the last one
! it would a ton!

what are the coordinates of the image of point w after rotating rectangle wxyz 270° counterclockwise about the origin? (see picture)
a. w’( -3, 4)
b. w’( 4, 3)
c. w’(3, 4)
d. w’(-3, -4)

D

Step-by-step explanation:

Under a counterclockwise rotation about the origin of 270°

a point (x, y ) →  (y, - x ), thus

W(4, - 3 ) → W'(- 3, - 4 ) → D

there are lots of ways.   pick's theorem tells us if the vertices are lattice points the area a is

a = b/2 + i - 1

where b is the number of boundary lattice points and i is the number of interior lattice points.

in the figure i count b=16, i=50 so

a = 8 + 50 - 1 = 57

let's try the shoelace formula.   the area is half the sum of the cross products of the sides.  

a(-2,6)   -2(2)-6(-5) = 26

b(-5,2)   -5(-3) - 2(-2) = 19

c(-2,-3) -2(-3) - (-3)(4) = 18

d(4,-3)   4(3) - (-3)(5) = 27

e(5,3)   5(3) - 3(1) = 12

f(1,3)     1(6) - 3(-2) = 12

a(-2,6)

each of the cross products is twice the area of the triangle with the associated side and one vertex the origin. so the area of abo=26/2=13.

area = (1/2) (26 + 19 +   18 + 27 + 12 + 12) = 57

that checks!

452,489 to the nearest thousand is 452,000