Question

question the mid - point of $overline{ab}$ is $m(1,0)$. if the coordinates of $a$ are $(6,7)$, what are the coordinates of $b$?

Explanation:

Step1: Recall mid - point formula

The mid - point formula for two points $A(x_1,y_1)$ and $B(x_2,y_2)$ is $M(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})$. Here $x_1 = 6,y_1 = 7$ and the mid - point $M(1,0)$.

Step2: Find the x - coordinate of B

Set up the equation for the x - coordinate of the mid - point: $\frac{x_1 + x_2}{2}=1$. Substitute $x_1 = 6$ into it: $\frac{6 + x_2}{2}=1$. Cross - multiply: $6 + x_2=2$. Solve for $x_2$: $x_2=2 - 6=-4$.

Step3: Find the y - coordinate of B

Set up the equation for the y - coordinate of the mid - point: $\frac{y_1 + y_2}{2}=0$. Substitute $y_1 = 7$ into it: $\frac{7 + y_2}{2}=0$. Cross - multiply: $7 + y_2=0$. Solve for $y_2$: $y_2=-7$.

Answer:

$(-4,-7)$