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Find the new triangle that undergoes a scale factor of 1/2 and a center of dilation at the origin a -7 , -4 b -3, 8 c 10 - 6Asked by 77
Find the new triangle that undergoes a scale factor of
1/2 and a center of dilation at the origin.
b=(-2,8)
a=(-8,-4)
c=(10,-6)
Answers
Answered by
GPT-4o mini
To find the vertices of the new triangle after applying a scale factor of \( \frac{1}{2} \) with the center of dilation at the origin, we will use the formula for dilation. The formula for a point \( (x, y) \) being dilated by a scale factor \( k \) from the origin is given by:
\[
(x', y') = (kx, ky)
\]
Given the points of the triangle:
- \( a = (-8, -4) \)
- \( b = (-2, 8) \)
- \( c = (10, -6) \)
We will apply the dilation to each vertex:
1. For point \( a \):
\[
a' = \left(\frac{1}{2} \cdot -8, \frac{1}{2} \cdot -4\right) = (-4, -2)
\]
2. For point \( b \):
\[
b' = \left(\frac{1}{2} \cdot -2, \frac{1}{2} \cdot 8\right) = (-1, 4)
\]
3. For point \( c \):
\[
c' = \left(\frac{1}{2} \cdot 10, \frac{1}{2} \cdot -6\right) = (5, -3)
\]
Thus, the new vertices of the triangle after the dilation are:
- \( a' = (-4, -2) \)
- \( b' = (-1, 4) \)
- \( c' = (5, -3) \)
These are the coordinates of the new triangle.
\[
(x', y') = (kx, ky)
\]
Given the points of the triangle:
- \( a = (-8, -4) \)
- \( b = (-2, 8) \)
- \( c = (10, -6) \)
We will apply the dilation to each vertex:
1. For point \( a \):
\[
a' = \left(\frac{1}{2} \cdot -8, \frac{1}{2} \cdot -4\right) = (-4, -2)
\]
2. For point \( b \):
\[
b' = \left(\frac{1}{2} \cdot -2, \frac{1}{2} \cdot 8\right) = (-1, 4)
\]
3. For point \( c \):
\[
c' = \left(\frac{1}{2} \cdot 10, \frac{1}{2} \cdot -6\right) = (5, -3)
\]
Thus, the new vertices of the triangle after the dilation are:
- \( a' = (-4, -2) \)
- \( b' = (-1, 4) \)
- \( c' = (5, -3) \)
These are the coordinates of the new triangle.
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