Question

the midpoint of \\(\overline{ab}\\) is \\(m(-2, -4)\\). if the coordinates of \\(a\\) are \\((-8, -5)\\), what are the coordinates of \\(b\\)?

Explanation:

Step1: Recall the midpoint formula

The midpoint \( M(x, y) \) of a line segment with endpoints \( A(x_1, y_1) \) and \( B(x_2, y_2) \) is given by \( x=\frac{x_1 + x_2}{2} \) and \( y=\frac{y_1 + y_2}{2} \).

Step2: Substitute the known values for the x - coordinate

We know that \( x=-2 \), \( x_1=-8 \). Substituting into the midpoint formula for the x - coordinate:
\( -2=\frac{-8 + x_2}{2} \)
Multiply both sides of the equation by 2: \( -2\times2=-8 + x_2 \)
\( -4=-8 + x_2 \)
Add 8 to both sides: \( x_2=-4 + 8=4 \)

Step3: Substitute the known values for the y - coordinate

We know that \( y = - 4 \), \( y_1=-5 \). Substituting into the midpoint formula for the y - coordinate:
\( -4=\frac{-5 + y_2}{2} \)
Multiply both sides of the equation by 2: \( -4\times2=-5 + y_2 \)
\( -8=-5 + y_2 \)
Add 5 to both sides: \( y_2=-8 + 5=-3 \)

Answer:

The coordinates of \( B \) are \( (4,-3) \)