Question

the midpoint of $overline{ab}$ is $m=(3, 1)$. one endpoint is $a=(8, 4)$. find the coordinates of the other endpoint, $b$. a (8, 4) m (3, 1) b (?,?) 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 \(A=(8,4)\), \(M=(3,1)\), and we want to find \(B=(x_2,y_2)\).

Step2: Solve for \(x_2\)

We know that \(\frac{x_1 + x_2}{2}=x_M\). Substituting \(x_1 = 8\) and \(x_M=3\), we get \(\frac{8 + x_2}{2}=3\). Cross - multiply: \(8 + x_2=6\), then \(x_2=6 - 8=-2\).

Step3: Solve for \(y_2\)

We know that \(\frac{y_1 + y_2}{2}=y_M\). Substituting \(y_1 = 4\) and \(y_M = 1\), we get \(\frac{4 + y_2}{2}=1\). Cross - multiply: \(4 + y_2=2\), then \(y_2=2 - 4=-2\).

Answer:

\((-2,-2)\)