Question14. Pricrice suppose a trilangle wros ditated by a seale factor of s whith cemter of ailations? and the image of that diation was oivated by a scale factor of it with center of slation still at , What single tranisformation would havis the same effect on the originad triangle? luselfy your answer with an harge.
Studdy Solution
STEP 1
1. The original triangle is dilated by a scale factor of with center of dilation at .
2. The resulting image is further dilated by a scale factor of with the same center .
3. We are to find a single transformation equivalent to these two successive dilations.
STEP 2
1. Understand the effect of successive dilations.
2. Derive the single transformation.
3. Justify the transformation mathematically.
STEP 3
Understand the effect of successive dilations:
When a figure is dilated by a scale factor of from a center , every point on the figure moves away from or towards by a factor of . If the resulting image is then dilated by another scale factor from the same center , each point on the image moves again by a factor of .
STEP 4
Derive the single transformation:
The overall effect of two successive dilations with scale factors and from the same center is equivalent to a single dilation with a scale factor equal to the product of the two scale factors. Therefore, the single transformation is a dilation with scale factor .
STEP 5
Justify the transformation mathematically:
Consider a point on the original triangle. After the first dilation, the image of is at , such that . After the second dilation, the image of is at , such that .
Substitute into the equation for :
This shows that the single transformation is a dilation with scale factor from the center .
The single transformation that has the same effect on the original triangle is a dilation with scale factor from the center .
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