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:
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For point \( a \): \[ a' = \left(\frac{1}{2} \cdot -8, \frac{1}{2} \cdot -4\right) = (-4, -2) \]
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For point \( b \): \[ b' = \left(\frac{1}{2} \cdot -2, \frac{1}{2} \cdot 8\right) = (-1, 4) \]
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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.