Describe the dilation of triangle ABC

Question

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, with a scale factor of 5 and a center point of dilation at the origin (0, 0)
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,

0
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. (1 point)
Responses

Triangle A′B′C′
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Answer 1

To describe the dilation of triangle $ABC$ with a scale factor of 5 and a center of dilation at the origin (0, 0), we can follow these steps:

1. Identify the vertices of triangle ABC: Let's assume the coordinates of the vertices of triangle $ABC$ are as follows:

   • $A(x_1,y_1)$
   • $B(x_2,y_2)$
   • $C(x_3,y_3)$

2. Apply the dilation formula: The formula for the dilation of a point $(x,y)$ with a scale factor $k$ about the origin is given by: $$(x^{\prime },y^{\prime })=(kx,ky)$$ In this case, our scale factor $k=5$.

3. Calculate the coordinates of the dilated triangle $A^{\prime }{B}^{\prime }{C}^{\prime }$:

   • The new coordinates for vertex $A^{\prime }$ will be: $$A^{\prime }=(5x_1,5y_1)$$
   • The new coordinates for vertex $B^{\prime }$ will be: $$B^{\prime }=(5x_2,5y_2)$$
   • The new coordinates for vertex $C^{\prime }$ will be: $$C^{\prime }=(5x_3,5y_3)$$

4. Conclusion: The new triangle formed by vertices $A^{\prime },B^{\prime },C^{\prime }$ after the dilation is denoted as triangle $A^{\prime }{B}^{\prime }{C}^{\prime }$. The coordinates of its vertices are five times the original coordinates of triangle $ABC$, effectively enlarging the triangle while maintaining its shape and proportional distances between the points.

In summary, triangle $A^{\prime }{B}^{\prime }{C}^{\prime }$ is the dilated version of triangle $ABC$ with vertices $A^{\prime }=(5x_1,5y_1)$, $B^{\prime }=(5x_2,5y_2)$, and $C^{\prime }=(5x_3,5y_3)$.