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The rule [tex]\( r_{y=x} \circ T_{4,0}(x, y) \)[/tex] is applied to trapezoid ABCD to produce the final image A"B"C"D".

Which ordered pairs are from the pre-image?

A. [tex]\((-1,0)\)[/tex]
B. [tex]\((-1,-5)\)[/tex]
C. [tex]\((1,1)\)[/tex]
D. [tex]\((7,0)\)[/tex]
E. [tex]\((7,-5)\)[/tex]

Sagot :

GinnyAnsweridn
Sure, let's go through each step of the transformation and reflection process step-by-step for each point of the trapezoid ABCD.

### Initial Points (Pre-Image)
The given vertices of the trapezoid are:
- [tex]\( A(-1, 0) \)[/tex]
- [tex]\( B(-1, -5) \)[/tex]
- [tex]\( C(1, 1) \)[/tex]
- [tex]\( D(7, 0) \)[/tex]
- [tex]\( E(7, -5) \)[/tex]

### Step 1: Translation by (4, 0)

When we translate each point by [tex]\( (4, 0) \)[/tex], we add 4 to the x-coordinate and keep the y-coordinate the same:

1. [tex]\( A(-1, 0) \to A'(-1 + 4, 0) = A'(3, 0) \)[/tex]
2. [tex]\( B(-1, -5) \to B'(-1 + 4, -5) = B'(3, -5) \)[/tex]
3. [tex]\( C(1, 1) \to C'(1 + 4, 1) = C'(5, 1) \)[/tex]
4. [tex]\( D(7, 0) \to D'(7 + 4, 0) = D'(11, 0) \)[/tex]
5. [tex]\( E(7, -5) \to E'(7 + 4, -5) = E'(11, -5) \)[/tex]

So after the translation, the points are:
- [tex]\( A'(3, 0) \)[/tex]
- [tex]\( B'(3, -5) \)[/tex]
- [tex]\( C'(5, 1) \)[/tex]
- [tex]\( D'(11, 0) \)[/tex]
- [tex]\( E'(11, -5) \)[/tex]

### Step 2: Reflection over the line [tex]\( y = x \)[/tex]

Next, we reflect each translated point over the line [tex]\( y = x \)[/tex]. This means swapping the x and y coordinates of each point:

1. [tex]\( A'(3, 0) \to A''(0, 3) \)[/tex]
2. [tex]\( B'(3, -5) \to B''(-5, 3) \)[/tex]
3. [tex]\( C'(5, 1) \to C''(1, 5) \)[/tex]
4. [tex]\( D'(11, 0) \to D''(0, 11) \)[/tex]
5. [tex]\( E'(11, -5) \to E''(-5, 11) \)[/tex]

So after the reflection, the final points are:
- [tex]\( A''(0, 3) \)[/tex]
- [tex]\( B''(-5, 3) \)[/tex]
- [tex]\( C''(1, 5) \)[/tex]
- [tex]\( D''(0, 11) \)[/tex]
- [tex]\( E''(-5, 11) \)[/tex]

### Summary

- Pre-image points: [tex]\((-1, 0)\)[/tex], [tex]\((-1, -5)\)[/tex], [tex]\((1, 1)\)[/tex], [tex]\((7, 0)\)[/tex], [tex]\((7, -5)\)[/tex]
- Translated points: [tex]\((3, 0)\)[/tex], [tex]\((3, -5)\)[/tex], [tex]\((5, 1)\)[/tex], [tex]\((11, 0)\)[/tex], [tex]\((11, -5)\)[/tex]
- Reflected points (Final image): [tex]\((0, 3)\)[/tex], [tex]\((-5, 3)\)[/tex], [tex]\((1, 5)\)[/tex], [tex]\((0, 11)\)[/tex], [tex]\((-5, 11)\)[/tex]

So, the ordered pairs for the initial image, translated image, and final reflected image of trapezoid ABCD are as detailed above.
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