A triangular pyramid is defined by the points .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.

Question

A triangular pyramid is defined by the points .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.

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. (Hint: Begin with the result of the rule for the reflection.)
Show the results of your calculations for the translation.

Any like advice that can help?

Answer 1

reflection: (x,y,z) -> (x,-y,z)

Now just apply the shifts.

Answer 2

To find the general rule for reflection, we need to understand how the reflection changes the coordinates of the points in the triangular pyramid.

For a reflection over the xz-plane, the y-coordinate of each point remains the same, while the x and z coordinates are negated. This means the general rule for the reflection can be written as follows:

(x, y, z) -> (-x, y, -z)

Now let's calculate the results of the reflection for the given points:

Let's assume the vertices of the triangular pyramid are A, B, C, and D. The coordinates for A, B, C, and D are (x₁, y₁, z₁), (x₂, y₂, z₂), (x₃, y₃, z₃), and (x₄, y₄, z₄) respectively.

After reflecting over the xz-plane, the new coordinates are:
A' = (-x₁, y₁, -z₁)
B' = (-x₂, y₂, -z₂)
C' = (-x₃, y₃, -z₃)
D' = (-x₄, y₄, -z₄)

Now let's move on to the translation.

For a translation, we simply add or subtract a constant value to each coordinate. In this case, the translation involves moving 3 units back, 2 units left, and 4 units up.

So, the general rule for translation can be written as:

(x, y, z) -> (x + Δx, y + Δy, z + Δz)

where Δx is the change in the x-coordinate (3 in this case), Δy is the change in the y-coordinate (0, as there is no change), and Δz is the change in the z-coordinate (4 in this case).

Calculating the results of the translation:

A'' = (-x₁ + 3, y₁ - 2, -z₁ + 4)
B'' = (-x₂ + 3, y₂ - 2, -z₂ + 4)
C'' = (-x₃ + 3, y₃ - 2, -z₃ + 4)
D'' = (-x₄ + 3, y₄ - 2, -z₄ + 4)

Note that the coordinates in the new system may differ depending on the specific values of (x₁, y₁, z₁), (x₂, y₂, z₂), (x₃, y₃, z₃), and (x₄, y₄, z₄) given for the triangular pyramid.

If you provide the specific values of the triangular pyramid's coordinates, I can help you calculate the results of the reflection and translation.