Question: User: C is the centre of rectangle DEFG.
Work out the coordinates of G.
D(?, ?), E(20, 25), C(8, 7), F(?, ?), G(?, ?)
Shape: rectangle DEFG centered at C(8, 7), with E at (20, 25) in the top-right area, D at the top-left, F at the bottom-right, and G at the bottom-left; position_hint: center
1. **State the problem:** We have a rectangle DEFG with center C at coordinates $C(8,7)$ and one vertex $E(20,25)$. We need to find the coordinates of vertex $G$.
2. **Recall the property of the center of a rectangle:** The center $C$ is the midpoint of the diagonal connecting opposite vertices. Here, $C$ is the midpoint of diagonal $EG$.
3. **Use the midpoint formula:** If $C$ is midpoint of $E(x_1,y_1)$ and $G(x_2,y_2)$, then
$$C_x = \frac{x_1 + x_2}{2}, \quad C_y = \frac{y_1 + y_2}{2}$$
4. **Substitute known values:**
$$8 = \frac{20 + x_G}{2}, \quad 7 = \frac{25 + y_G}{2}$$
5. **Solve for $x_G$ and $y_G$:**
Multiply both sides by 2:
$$16 = 20 + x_G, \quad 14 = 25 + y_G$$
Apply cancellation:
$$16 = \cancel{20} + x_G \Rightarrow x_G = 16 - 20 = -4$$
$$14 = \cancel{25} + y_G \Rightarrow y_G = 14 - 25 = -11$$
6. **Final answer:**
$$G(-4, -11)$$