Mariam T-yan's Question on Facebook (Reflections of Pyramids)

[Sudharaka]

Mariam T-yan of Facebook writes:

Please help me so I can graduate

1.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. (Hint: Begin with the result of the rule for the reflection.)

• Show the results of your calculations for the translation.

 

[jvicens]

Hi Mariam,

I can help you with this problem. First, let's start by writing the general rule for reflection over the xz-plane. The x-coordinate will remain the same, the y-coordinate will become its negative, and the z-coordinate will remain the same. So, the general rule for reflection over the xz-plane is (x, y, z) --> (x, -y, z).

Now, let's apply this rule to the given coordinates. The reflected image will have the following coordinates:

P'(0, -5, 4)
Y'(-2, -7, 4)
R'(0, -7, 4)
A'(0, -7, 6)

Next, we need to apply the translation of 3 units back, 2 units left, and 4 units up. This translation will change the x-coordinate by -3, the y-coordinate by -2, and the z-coordinate by 4. So, the general rule for translation is (x, y, z) --> (x-3, y-2, z+4).

Applying this rule to the reflected coordinates, we get the final image coordinates:

P''(-3, -7, 8)
Y''(-5, -9, 8)
R''(-3, -9, 8)
A''(-3, -9, 10)

Therefore, the final image coordinates are P''(-3, -7, 8), Y''(-5, -9, 8), R''(-3, -9, 8), and A''(-3, -9, 10).

I hope this helps you with your problem. Let me know if you have any further questions.