Question

Triangle ABC has a vertices point A ( -2 , -2), point B (-4 , -3), and point C (-2 , -5). Find the coordinates of A' B' and C' after a dilation with a scale factor of 3 and a center point of dilation at the origin.

Answer 1

To find the coordinates of the points A', B', and C' after a dilation with a scale factor of 3 from the origin, we can use the formula for dilation:

\[
(x', y') = (kx, ky)
\]

where \((x, y)\) are the original coordinates of the point, \(k\) is the scale factor, and \((x', y')\) are the coordinates after dilation.

Given the scale factor \(k = 3\) and the center of dilation at the origin \((0, 0)\), we will apply the formula to each vertex of triangle ABC.

1. For point A \((-2, -2)\):

\[
A' = (3 \cdot -2, 3 \cdot -2) = (-6, -6)
\]

2. For point B \((-4, -3)\):

\[
B' = (3 \cdot -4, 3 \cdot -3) = (-12, -9)
\]

3. For point C \((-2, -5)\):

\[
C' = (3 \cdot -2, 3 \cdot -5) = (-6, -15)
\]

Thus, the coordinates after the dilation are:

- \(A' = (-6, -6)\)
- \(B' = (-12, -9)\)
- \(C' = (-6, -15)\)

Therefore, the coordinates of the points A', B', and C' after the dilation are:

\[
A'(-6, -6), \quad B'(-12, -9), \quad C'(-6, -15)
\]