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  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.

´Show the result of your calculations for the translation.

Verified Answer :

Edufirst

General rule for the reflection. r(x,y,z)= f(x,-y,z)

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

General rule for the translation: t(x,y,z) = r(x+2, y-4,z+3)

Result:

P: (0-2, -5+4 , 4-3) = (-2, -1, 1)
Y: (-2-2, - 7+4, 4-3) = (-4, -3, 1)
R: (0-2, - 7+4, 4-3) = (-2, -3, 1)
A: (0-2, - 7+4, 6-3) = (-2, -3, 3)


 

 

Answer:

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).