Question

5.
ΔABC has coordinates of A (–6, –3), B (7, –7), and C (0, 3). Find the coordinates of its image after a dilation centered at the origin with a scale factor of 1.5.

Answer 1

To find the coordinates of the image of triangle ΔABC after a dilation centered at the origin with a scale factor of 1.5, we apply the dilation transformation to each vertex of the triangle. The formula for dilation about the origin with a scale factor \( k \) is given by:

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

where \( (x, y) \) are the original coordinates of the point, and \( (x', y') \) are the new coordinates after dilation.

Now, we will apply this formula to each vertex of triangle ABC.

1. For point A (–6, –3):
\[
A' = (1.5 \cdot -6, 1.5 \cdot -3) = (-9, -4.5)
\]

2. For point B (7, –7):
\[
B' = (1.5 \cdot 7, 1.5 \cdot -7) = (10.5, -10.5)
\]

3. For point C (0, 3):
\[
C' = (1.5 \cdot 0, 1.5 \cdot 3) = (0, 4.5)
\]

After the dilation, the coordinates of the image of triangle ΔABC are:

- A' = (-9, -4.5)
- B' = (10.5, -10.5)
- C' = (0, 4.5)

Therefore, the final coordinates of the image after the dilation are:

\[
A' (-9, -4.5), \quad B' (10.5, -10.5), \quad C' (0, 4.5)
\]