Question: Here is triangle ABC.
Write an equation for
the line containing all
possible images of
point C after a dilation
with center (2, 0).
Triangle ABC is shown on a coordinate grid in the center-right; A is at (2, 0), B is at (5, 0), and C is at (5, 1), with a segment from A to C and a vertical segment from B to C.
1. **State the problem:** We need to find the equation of the line containing all possible images of point $C(5,1)$ after a dilation centered at $O(2,0)$ with any scale factor.
2. **Recall the dilation formula:** A dilation with center $O(x_0,y_0)$ and scale factor $k$ maps a point $P(x,y)$ to $P'(x',y')$ given by:
$$
x' = x_0 + k(x - x_0), \quad y' = y_0 + k(y - y_0)
$$
3. **Apply the formula to point $C(5,1)$:**
$$
x' = 2 + k(5 - 2) = 2 + 3k
$$
$$
y' = 0 + k(1 - 0) = k
$$
4. **Express $k$ from $y'$:**
$$
k = y'
$$
Substitute into $x'$:
$$
x' = 2 + 3y'
$$
5. **Rewrite as an equation of a line:**
$$
x' - 3y' = 2
$$
Or equivalently:
$$
x' = 3y' + 2
$$
6. **Interpretation:** The set of all images of $C$ under dilation with center $(2,0)$ lies on the line $x = 3y + 2$.
**Final answer:**
$$
x = 3y + 2
$$