2. a triangular pyramid begins with these coordinates:
p: (0, 5, 4)
y: (_2, 7, 4)
r: (0, 7, 4)
a: (0, 7, 6)
the pyramid is then reflected over the xz-plane. the reflected image is then translated 3 units back, 2 units left, and 4 units up. determine the coordinates of that final image, following these steps:
´ write the general rule you use for the reflection.
´ show the results of your calculations for the reflection.
´ write the general rule you use for the translation.
´show the result of your calculations for the translation.

1)

(x,y,z) → (x,-y,z).

2)

P:(0,5,4) → P'(0,-5,4)

Y: (-2, 7, 4)  → Y'(-2,-7,4)

R: (0, 7, 4)  → R'(0,-7,4)

A: (0, 7, 6) → A'(0,-7,6)

3)

(x',y',z') → (x'-3,y'-2,z'+4).

4)

P:(0,5,4) → P"(-3,-7,8)

Y: (-2, 7, 4)  → Y"(-5,-9,8)

R: (0, 7, 4)  → R"(-3,-9,8)

A: (0, 7, 6) → A"(-3,-9,10).

Step-by-step explanation:

A triangular pyramid begins with these coordinates:

P: (0, 5, 4)

Y: (-2, 7, 4)

R: (0, 7, 4)

A: (0, 7, 6)

1)

The pyramid is reflected over the xz-plane.

so the rule of the reflection will be:

The points x,y,z are transformed to x,-y,z.

i.e. (x,y,z) → (x',y',z').

where (x'y'z')=(x,-y,z)

i.e. (x,y,z) → (x,-y,z)

2)

The coordinates of the points after reflection is:

P:(0,5,4) → P'(0,-5,4)

Y: (-2, 7, 4)  → Y'(-2,-7,4)

R: (0, 7, 4)  → R'(0,-7,4)

A: (0, 7, 6) → A'(0,-7,6)

3)

The general rule of translation is:

(x',y',z') → (x'-3,y'-2,z'+4).

4)

Hence, our point P,Y,R and A are given as:

P:(0,5,4) → P''(0-3,-5-2,4+4)=P"(-3,-7,8)

Y: (-2, 7, 4)  → Y''(-2-3,-7-2,4+4)=Y"(-5,-9,8)

R: (0, 7, 4)  → R''(0-3,-7-2,4+4)=R"(-3,-9,8)

A: (0, 7, 6) → A''(0-3,-7-2,6+4)=A"(-3,-9,10).

i.e.

P:(0,5,4) → P"(-3,-7,8)

Y: (-2, 7, 4)  → Y"(-5,-9,8)

R: (0, 7, 4)  → R"(-3,-9,8)

A: (0, 7, 6) → A"(-3,-9,10).

2

step-by-step explanation: