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Describe the dilation of triangle ABC

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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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Responses

Triangle A′B′C′
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1 answer

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: \[ (x', y') = (kx, ky) \] In this case, our scale factor \( k = 5 \).

  3. Calculate the coordinates of the dilated triangle \( A'B'C' \):

    • The new coordinates for vertex \( A' \) will be: \[ A' = (5x_1, 5y_1) \]
    • The new coordinates for vertex \( B' \) will be: \[ B' = (5x_2, 5y_2) \]
    • The new coordinates for vertex \( C' \) will be: \[ C' = (5x_3, 5y_3) \]

  4. Conclusion: The new triangle formed by vertices \( A', B', C' \) after the dilation is denoted as triangle \( A'B'C' \). 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 \( A'B'C' \) is the dilated version of triangle \( ABC \) with vertices \( A' = (5x_1, 5y_1) \), \( B' = (5x_2, 5y_2) \), and \( C' = (5x_3, 5y_3) \).