Question

point $m$ is the midpoint of point $a$ and point $b$. find the coordinates of point $b$.
$a(1,10)$
$m\left(\frac{23}{2},0\
ight)$

Explanation:

Step1: Recall midpoint formula

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

Step2: Solve for $x_b$

Substitute $x_m=\frac{23}{2}$, $x_a=1$ into the x-coordinate formula:
$$\frac{23}{2} = \frac{1 + x_b}{2}$$
Multiply both sides by 2: $23 = 1 + x_b$
Solve for $x_b$: $x_b = 23 - 1 = 22$

Step3: Solve for $y_b$

Substitute $y_m=0$, $y_a=10$ into the y-coordinate formula:
$$0 = \frac{10 + y_b}{2}$$
Multiply both sides by 2: $0 = 10 + y_b$
Solve for $y_b$: $y_b = 0 - 10 = -10$

Answer:

$(22, -10)$