Question

the midpoint of \\( \overline { a b } \\) is \\( m ( - 4,4 ) \\). if the coordinates of \\( a \\) are \\( ( - 3,1 ) \\), what are the coordinates of \\( b \\)

Explanation:

Step1: Use mid - point formula

The mid - point formula for two points \(A(x_1,y_1)\) and \(B(x_2,y_2)\) is \(M(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})\). Let \(A(-3,1)\) and \(M(-4,4)\), and \(B=(x,y)\).
So, \(\frac{-3 + x}{2}=-4\) and \(\frac{1 + y}{2}=4\).

Step2: Solve for \(x\)

For \(\frac{-3 + x}{2}=-4\), multiply both sides by \(2\): \(-3 + x=-8\). Then add \(3\) to both sides: \(x=-8 + 3=-5\).

Step3: Solve for \(y\)

For \(\frac{1 + y}{2}=4\), multiply both sides by \(2\): \(1 + y = 8\). Then subtract \(1\) from both sides: \(y=8 - 1 = 7\).

Answer:

The coordinates of \(B\) are \((-5,7)\)