To solve this problem, we are given:

• The midpoint of $\overline{AB}$ is at $(0, 0)$,
• The coordinates of point $B$ are $(2, -3)$.

We are asked to find the coordinates of point $A$.

Using the Midpoint Formula:

The formula for the midpoint between two points $A(x_1, y_1)$ and $B(x_2, y_2)$ is:

$M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$

Here, the midpoint $M$ is $(0, 0)$ and the coordinates of point $B$ are $(2, -3)$. Let the coordinates of point $A$ be $(x_1, y_1)$.

We can set up two equations using the midpoint formula:

1. For the x-coordinate: $\frac{x_1 + 2}{2} = 0$ Multiplying both sides by 2: $x_1 + 2 = 0$ Solving for $x_1$: $x_1 = -2$

2. For the y-coordinate: $\frac{y_1 - 3}{2} = 0$ Multiplying both sides by 2: $y_1 - 3 = 0$ Solving for $y_1$: $y_1 = 3$

Therefore, the coordinates of point $A$ are $(-2, 3)$.

Would you like further clarification, or have any questions?

Related Questions:

1. What is the formula for the distance between two points?
2. How would you find the midpoint if you were given both points $A$ and $B$?
3. How can you verify the midpoint of two points?
4. If the midpoint was shifted to a different location, how would that affect the coordinates of $A$?
5. What happens to the midpoint if $A$ and $B$ are on the same axis?

Tip:

Always double-check the midpoint formula by plugging the values back in to ensure accuracy!