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Sagot :
Certainly! Let's walk through the process of translating each given ordered pair by the translation vector [tex]\( T(4, -3) \)[/tex].
### Explanation of Translation:
A translation moves every point of a figure or a space by the same amount in a given direction. In coordinate geometry, if you want to translate a point [tex]\((x, y)\)[/tex] by a vector [tex]\( (a, b) \)[/tex], you add [tex]\( a \)[/tex] to the [tex]\( x \)[/tex]-coordinate and [tex]\( b \)[/tex] to the [tex]\( y \)[/tex]-coordinate. This means the translated point will be [tex]\((x + a, y + b)\)[/tex].
For this problem, the translation vector given is [tex]\( (4, -3) \)[/tex]. This means we will add 4 to the x-coordinate and subtract 3 from the y-coordinate for each pair.
### Given Ordered Pairs and Translation Process:
1. Original Pair: [tex]\( (-9, 2) \)[/tex]
- Translated X-coordinate: [tex]\( -9 + 4 = -5 \)[/tex]
- Translated Y-coordinate: [tex]\( 2 - 3 = -1 \)[/tex]
- Translated Pair: [tex]\( (-5, -1) \)[/tex]
2. Original Pair: [tex]\( (0, -5) \)[/tex]
- Translated X-coordinate: [tex]\( 0 + 4 = 4 \)[/tex]
- Translated Y-coordinate: [tex]\( -5 - 3 = -8 \)[/tex]
- Translated Pair: [tex]\( (4, -8) \)[/tex]
3. Original Pair: [tex]\( (-5, -1) \)[/tex]
- Translated X-coordinate: [tex]\( -5 + 4 = -1 \)[/tex]
- Translated Y-coordinate: [tex]\( -1 - 3 = -4 \)[/tex]
- Translated Pair: [tex]\( (-1, -4) \)[/tex]
4. Original Pair: [tex]\( (12, 5) \)[/tex]
- Translated X-coordinate: [tex]\( 12 + 4 = 16 \)[/tex]
- Translated Y-coordinate: [tex]\( 5 - 3 = 2 \)[/tex]
- Translated Pair: [tex]\( (16, 2) \)[/tex]
5. Original Pair: [tex]\( (-5, -1) \)[/tex] (repeated)
- Translated X-coordinate: [tex]\( -5 + 4 = -1 \)[/tex]
- Translated Y-coordinate: [tex]\( -1 - 3 = -4 \)[/tex]
- Translated Pair: [tex]\( (-1, -4) \)[/tex] (same as before)
6. Original Pair: [tex]\( (4, -8) \)[/tex]
- Translated X-coordinate: [tex]\( 4 + 4 = 8 \)[/tex]
- Translated Y-coordinate: [tex]\( -8 - 3 = -11 \)[/tex]
- Translated Pair: [tex]\( (8, -11) \)[/tex]
7. Original Pair: [tex]\( (5, 2) \)[/tex]
- Translated X-coordinate: [tex]\( 5 + 4 = 9 \)[/tex]
- Translated Y-coordinate: [tex]\( 2 - 3 = -1 \)[/tex]
- Translated Pair: [tex]\( (9, -1) \)[/tex]
8. Original Pair: [tex]\( (6, 9) \)[/tex]
- Translated X-coordinate: [tex]\( 6 + 4 = 10 \)[/tex]
- Translated Y-coordinate: [tex]\( 9 - 3 = 6 \)[/tex]
- Translated Pair: [tex]\( (10, 6) \)[/tex]
9. Original Pair: [tex]\( (5, -2) \)[/tex]
- Translated X-coordinate: [tex]\( 5 + 4 = 9 \)[/tex]
- Translated Y-coordinate: [tex]\( -2 - 3 = -5 \)[/tex]
- Translated Pair: [tex]\( (9, -5) \)[/tex]
10. Original Pair: [tex]\( (1, 1) \)[/tex]
- Translated X-coordinate: [tex]\( 1 + 4 = 5 \)[/tex]
- Translated Y-coordinate: [tex]\( 1 - 3 = -2 \)[/tex]
- Translated Pair: [tex]\( (5, -2) \)[/tex]
### Summary of Translated Pairs:
After applying the translation [tex]\( T(4, -3) \)[/tex] to each given pair, the resulting translated pairs are:
[tex]\[ (-5, -1), (4, -8), (-1, -4), (16, 2), (-1, -4), (8, -11), (9, -1), (10, 6), (9, -5), (5, -2) \][/tex]
These are the new coordinates of the points after the translation.
### Explanation of Translation:
A translation moves every point of a figure or a space by the same amount in a given direction. In coordinate geometry, if you want to translate a point [tex]\((x, y)\)[/tex] by a vector [tex]\( (a, b) \)[/tex], you add [tex]\( a \)[/tex] to the [tex]\( x \)[/tex]-coordinate and [tex]\( b \)[/tex] to the [tex]\( y \)[/tex]-coordinate. This means the translated point will be [tex]\((x + a, y + b)\)[/tex].
For this problem, the translation vector given is [tex]\( (4, -3) \)[/tex]. This means we will add 4 to the x-coordinate and subtract 3 from the y-coordinate for each pair.
### Given Ordered Pairs and Translation Process:
1. Original Pair: [tex]\( (-9, 2) \)[/tex]
- Translated X-coordinate: [tex]\( -9 + 4 = -5 \)[/tex]
- Translated Y-coordinate: [tex]\( 2 - 3 = -1 \)[/tex]
- Translated Pair: [tex]\( (-5, -1) \)[/tex]
2. Original Pair: [tex]\( (0, -5) \)[/tex]
- Translated X-coordinate: [tex]\( 0 + 4 = 4 \)[/tex]
- Translated Y-coordinate: [tex]\( -5 - 3 = -8 \)[/tex]
- Translated Pair: [tex]\( (4, -8) \)[/tex]
3. Original Pair: [tex]\( (-5, -1) \)[/tex]
- Translated X-coordinate: [tex]\( -5 + 4 = -1 \)[/tex]
- Translated Y-coordinate: [tex]\( -1 - 3 = -4 \)[/tex]
- Translated Pair: [tex]\( (-1, -4) \)[/tex]
4. Original Pair: [tex]\( (12, 5) \)[/tex]
- Translated X-coordinate: [tex]\( 12 + 4 = 16 \)[/tex]
- Translated Y-coordinate: [tex]\( 5 - 3 = 2 \)[/tex]
- Translated Pair: [tex]\( (16, 2) \)[/tex]
5. Original Pair: [tex]\( (-5, -1) \)[/tex] (repeated)
- Translated X-coordinate: [tex]\( -5 + 4 = -1 \)[/tex]
- Translated Y-coordinate: [tex]\( -1 - 3 = -4 \)[/tex]
- Translated Pair: [tex]\( (-1, -4) \)[/tex] (same as before)
6. Original Pair: [tex]\( (4, -8) \)[/tex]
- Translated X-coordinate: [tex]\( 4 + 4 = 8 \)[/tex]
- Translated Y-coordinate: [tex]\( -8 - 3 = -11 \)[/tex]
- Translated Pair: [tex]\( (8, -11) \)[/tex]
7. Original Pair: [tex]\( (5, 2) \)[/tex]
- Translated X-coordinate: [tex]\( 5 + 4 = 9 \)[/tex]
- Translated Y-coordinate: [tex]\( 2 - 3 = -1 \)[/tex]
- Translated Pair: [tex]\( (9, -1) \)[/tex]
8. Original Pair: [tex]\( (6, 9) \)[/tex]
- Translated X-coordinate: [tex]\( 6 + 4 = 10 \)[/tex]
- Translated Y-coordinate: [tex]\( 9 - 3 = 6 \)[/tex]
- Translated Pair: [tex]\( (10, 6) \)[/tex]
9. Original Pair: [tex]\( (5, -2) \)[/tex]
- Translated X-coordinate: [tex]\( 5 + 4 = 9 \)[/tex]
- Translated Y-coordinate: [tex]\( -2 - 3 = -5 \)[/tex]
- Translated Pair: [tex]\( (9, -5) \)[/tex]
10. Original Pair: [tex]\( (1, 1) \)[/tex]
- Translated X-coordinate: [tex]\( 1 + 4 = 5 \)[/tex]
- Translated Y-coordinate: [tex]\( 1 - 3 = -2 \)[/tex]
- Translated Pair: [tex]\( (5, -2) \)[/tex]
### Summary of Translated Pairs:
After applying the translation [tex]\( T(4, -3) \)[/tex] to each given pair, the resulting translated pairs are:
[tex]\[ (-5, -1), (4, -8), (-1, -4), (16, 2), (-1, -4), (8, -11), (9, -1), (10, 6), (9, -5), (5, -2) \][/tex]
These are the new coordinates of the points after the translation.
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