1. a rectangular prism is defined as the drawing shows. note b at (0, 5, 0) and f (0, 0, 4). the prism is reflected over the xz-plane. the reflected image is then translated 2 units back, 3 units left, and 1 unit up.
a.) show the results of your calculations for points a, b, c and d from the reflection.
b.) show the results of your calculations for points a, b, c and d from the translation.

Initial coordinates:
A (0,0,0)
B (0,5,0)
C (3,5,0)
D (3,0,0)
E (3,0,4)
F (0,0,4)
G (0,5,4)
H (3,5,4)

a.) The prism is reflected over the xz-plane.
Reflection Point P(x,y,z) over the xz-plane is P'(x,-y,z):
Reflection P(x,y,z) over the xz-plane → P'(x,-y,z)

Reflection A(0,0,0) over the xz-plane → A'(0,-0,0) = A'(0,0,0)
Reflection B(0,5,0) over the xz-plane → B'(0,-5,0)
Reflection C(3,5,0) over the xz-plane → C'(3,-5,0)
Reflection D(3,0,0) over the xz-plane → D'(3,-0,0) = D'(3,0,0)

b.) The reflected image is then translated 2 units back, 3 units left, and 1 unit up.
Translation P'(x,-y,z) 2 units back, 3 units left, and 1 unit up is P''(x-2,-y-3,z+1):
Translation P'(x,-y,z) → P''(x-2,-y-3,z+1)

Translation A'(0,0,0) →A''(0-2,0-3,0+1) = A''(-2,-3,1)
Translation B'(0,-5,0) →B''(0-2,-5-3,0+1) = B''(-2,-8,1)
Translation C'(3,-5,0) →C''(3-2,-5-3,0+1) = C''(1,-8,1)
Translation D'(3,0,0) →D''(3-2,0-3,0+1) = D''(1,-3,1)

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