Question

1. the midpoint of \\( \overline { a b } \\) is \\( m ( 4,2 ) \\). if the coordinates of \\( a \\) are \\( ( 6, - 4 ) \\), what are the coordinates of \\( b \\)?

a) \\( ( 1, - 3 ) \\)
b) \\( ( 2,8 ) \\)
c) \\( ( 5, - 1 ) \\)
d) \\( ( 14,0 ) \\)

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 \(M(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})\). Given \(M(4,2)\) and \(A(6,-4)\), let \(B=(x,y)\). Then \(\frac{6 + x}{2}=4\) and \(\frac{-4 + y}{2}=2\).

Step2: Solve for \(x\)

For \(\frac{6 + x}{2}=4\), multiply both sides by \(2\): \(6 + x = 8\). Subtract \(6\) from both sides: \(x=8 - 6=2\).

Step3: Solve for \(y\)

For \(\frac{-4 + y}{2}=2\), multiply both sides by \(2\): \(-4 + y = 4\). Add \(4\) to both sides: \(y=4 + 4 = 8\).

Answer:

B. \((2,8)\)