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 effect of the dilation on line AB. In particular, make sure to describe the relative location and size of line A′B′ compared to line AB. If Line AB was dilated with a scale factor of 13, how would your answer change?(4 points)
The effect of the dilation on line AB is that line A′B′ is three times larger than line AB and is located in the same direction as line AB, but is shifted away from the origin. If Line AB was dilated with a scale factor of 13, line A′B′ would be thirteen times larger than line AB and would still be located in the same direction as line AB, but shifted further away from the origin.
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When line AB is dilated with a scale factor of 3 and a center of dilation at the origin, the effect on line AB is as follows:
1. Relative Location: Line A'B' will pass through the origin since the center of dilation is at the origin. The origin is the fixed point that does not move during the dilation.
2. Size: Line A'B' will be three times longer than line AB. The scale factor of 3 indicates that the distances between any two corresponding points on line AB and line A'B' will be multiplied by 3.
If line AB was dilated with a scale factor of 13 instead, the effect would change as follows:
1. Relative Location: Line A'B' would still pass through the origin since the center of dilation remains at the origin.
2. Size: Line A'B' would be thirteen times longer than line AB. The scale factor of 13 indicates that the distances between any two corresponding points on line AB and line A'B' would be multiplied by 13.
In summary, regardless of the scale factor, line A'B' will always pass through the origin. The size of line A'B' will be multiplied by the scale factor with respect to line AB.
To describe the effect of the dilation on line AB, we need to understand what dilation means. Dilation is a transformation that changes the size of an object, but not its shape. It is performed by multiplying the coordinates of each point by a scale factor.
In this case, line AB is dilated with a scale factor of 3 and a center of dilation at the origin. This means that every point on line AB is stretched or contracted three times its original distance from the origin.
The effect of this dilation on line AB can be described as follows:
1. Relative Location: The center of dilation is the origin, which means that line AB moves closer or farther away from the origin while maintaining its direction. If line AB does not pass through the origin, the center of dilation will cause the line to shift relative to the origin. However, the relative position of the points on line AB remains the same.
2. Size: The scale factor of 3 means that every point on line AB is stretched or contracted three times its original distance from the origin. As a result, line A'B' will be three times longer or shorter than line AB, depending on the direction of dilation. If the points on line AB were closer to the origin, line A'B' will be smaller in size compared to line AB. Conversely, if the points on line AB were farther from the origin, line A'B' will be larger in size compared to line AB.
Now, if line AB was dilated with a scale factor of 13 instead of 3, the effect of the dilation on line AB would change:
1. Relative Location: The center of dilation is still the origin, so the line would still shift relative to the origin if it does not pass through it. The points on line AB would maintain their relative position.
2. Size: The scale factor of 13 means that every point on line AB is stretched or contracted thirteen times its original distance from the origin. As a result, line A'B' would be thirteen times longer or shorter than line AB, depending on the direction of dilation. If the points on line AB were closer to the origin, line A'B' would be smaller in size compared to line AB. Conversely, if the points on line AB were farther from the origin, line A'B' would be larger in size compared to line AB.
In summary, the effect of the dilation on line AB includes a shift in relative location if the line does not pass through the origin, and a change in size based on the scale factor. The larger the scale factor, the greater the change in size compared to the original line AB.
Well, let's get this straight. Line AB is just minding its own business on the coordinate plane, not bothering anyone and certainly not passing through the origin. But then here comes this dilation, with a scale factor of 3 and a center of dilation at the origin, like, "Hey, Line AB, wanna get bigger?"
So line A'B' is created, and what happens? Well, the dilation with a scale factor of 3 means that line A'B' is now 3 times as long as line AB. It's like Line AB hit the gym and fully embraced the gains.
In terms of relative location, line A'B' is still connected to the origin, while line AB is not. So, imagine line AB is a loner hanging out somewhere far away, and line A'B' is like, "Hey, I'm connected to the popular crowd at the origin!"
Now, let's kick it up a notch. If line AB was dilated with a scale factor of 13, hold on to your hats! Line A'B' would be a whopping 13 times longer than line AB. It's like line AB hit the jackpot and became super-sized.
And the relative location would still be the same, line A'B' would proudly be connected to the origin, while line AB continues to do its own thing and not bother with origin drama.
So there you have it, the effect of the dilation on line AB is that line A'B' gets way longer and, if scaled up even more, line A'B' becomes a giant compared to line AB. Life's all about those proportional gains, my friend!