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Line AB is on the coordinate plane and does not pass through the origin. Line AB is dilated with a scale factor of 3 and a cent...Question
Line AB is on the coordinate plane and does not pass through the origin. Line AB is dilated with a scale factor of 3 and a center of dilation at the origin to create line A′B′ . Describe the effects of the dilation on Line AB . If line AB was dilated with a scale factor of 13 , how would your answer change?(4 points)
human short for a 6th grader
human short for a 6th grader
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Answered by
GPT-4o mini
When we dilate line AB with a scale factor of 3 and the center of dilation at the origin, we are basically making the line longer and farther away from the origin.
For example, every point on line AB is moved away from the origin by a factor of 3. If a point on line AB is at coordinates (x, y), after the dilation, its new position (A′) will be at (3x, 3y). This makes the line A′B′ three times larger and keeps it parallel to the original line AB. Also, because the center of dilation is at the origin, line A′B′ will still not pass through the origin.
If we dilate line AB with a scale factor of 13 instead, the same thing happens, but each point on the line is moved even farther away from the origin. After dilation, a point that was at (x, y) becomes (13x, 13y). This means line A′B′ will be 13 times larger than the original line AB and still parallel to it.
In summary:
- With a scale factor of 3, the line is 3 times larger and still parallel, not going through the origin.
- With a scale factor of 13, the line is 13 times larger and still parallel, also not going through the origin. The larger the scale factor, the farther from the origin the new line will be!
For example, every point on line AB is moved away from the origin by a factor of 3. If a point on line AB is at coordinates (x, y), after the dilation, its new position (A′) will be at (3x, 3y). This makes the line A′B′ three times larger and keeps it parallel to the original line AB. Also, because the center of dilation is at the origin, line A′B′ will still not pass through the origin.
If we dilate line AB with a scale factor of 13 instead, the same thing happens, but each point on the line is moved even farther away from the origin. After dilation, a point that was at (x, y) becomes (13x, 13y). This means line A′B′ will be 13 times larger than the original line AB and still parallel to it.
In summary:
- With a scale factor of 3, the line is 3 times larger and still parallel, not going through the origin.
- With a scale factor of 13, the line is 13 times larger and still parallel, also not going through the origin. The larger the scale factor, the farther from the origin the new line will be!
Answered by
Claude 3.5 Sonnet
Let me explain the effects of dilation in simple terms for a 6th grader:
For scale factor of 3:
- Line A'B' will be 3 times longer than line AB
- Line A'B' will point in the same direction as AB
- Each point on line A'B' will be 3 times farther from the origin compared to the matching point on AB
For scale factor of 1/3:
- It's the opposite! Line A'B' will be 1/3 as long as line AB
- The direction stays the same
- Each point on A'B' will be 1/3 as far from the origin compared to AB
Think of it like zooming in (scale factor 3) or zooming out (scale factor 1/3) with the origin as your center point.
For scale factor of 3:
- Line A'B' will be 3 times longer than line AB
- Line A'B' will point in the same direction as AB
- Each point on line A'B' will be 3 times farther from the origin compared to the matching point on AB
For scale factor of 1/3:
- It's the opposite! Line A'B' will be 1/3 as long as line AB
- The direction stays the same
- Each point on A'B' will be 1/3 as far from the origin compared to AB
Think of it like zooming in (scale factor 3) or zooming out (scale factor 1/3) with the origin as your center point.
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