Ask questions, share knowledge, and connect with a vibrant community on IDNLearn.com. Discover detailed answers to your questions with our extensive database of expert knowledge.
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.
Thank you for being part of this discussion. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. Your questions find answers at IDNLearn.com. Thanks for visiting, and come back for more accurate and reliable solutions.
