Question

question
the midpoint of \\( \overline{ab} \\) is \\( m(-1, -1) \\). if the coordinates of \\( a \\) are \\( (1, 2) \\), what are the coordinates of \\( b \\)?
answer

Explanation:

Step1: Recall midpoint formula

The midpoint $M(x_m, y_m)$ of a segment with endpoints $A(x_a, y_a)$ and $B(x_b, y_b)$ is given by:
$$x_m = \frac{x_a + x_b}{2}, \quad y_m = \frac{y_a + y_b}{2}$$

Step2: Substitute known x-values

We know $x_m=-1$, $x_a=1$. Solve for $x_b$:
$$-1 = \frac{1 + x_b}{2}$$
Multiply both sides by 2: $-2 = 1 + x_b$
Subtract 1: $x_b = -2 - 1 = -3$

Step3: Substitute known y-values

We know $y_m=-1$, $y_a=2$. Solve for $y_b$:
$$-1 = \frac{2 + y_b}{2}$$
Multiply both sides by 2: $-2 = 2 + y_b$
Subtract 2: $y_b = -2 - 2 = -4$

Answer:

The coordinates of $B$ are $(-3, -4)$