Question

question
the midpoint of \\(\overline{ab}\\) is \\(m(-4, 0)\\). if the coordinates of \\(a\\) are \\((-2, -8)\\), what are the coordinates of \\(b\\)?

Explanation:

Step1: Recall midpoint formula

The midpoint \( M(x_m, y_m) \) of a line segment with endpoints \( A(x_1, y_1) \) and \( B(x_2, y_2) \) is given by \( x_m=\frac{x_1 + x_2}{2} \) and \( y_m=\frac{y_1 + y_2}{2} \). We know \( M(-4,0) \) and \( A(-2,-8) \), so \( x_1=-2 \), \( y_1 = -8 \), \( x_m=-4 \), \( y_m = 0 \). We need to find \( x_2 \) and \( y_2 \) (coordinates of \( B \)).

Step2: Solve for \( x_2 \)

Using the x - coordinate formula of midpoint: \( -4=\frac{-2 + x_2}{2} \). Multiply both sides by 2: \( -4\times2=-2 + x_2 \), so \( -8=-2 + x_2 \). Add 2 to both sides: \( x_2=-8 + 2=-6 \).

Step3: Solve for \( y_2 \)

Using the y - coordinate formula of midpoint: \( 0=\frac{-8 + y_2}{2} \). Multiply both sides by 2: \( 0\times2=-8 + y_2 \), so \( 0=-8 + y_2 \). Add 8 to both sides: \( y_2=0 + 8 = 8 \).

Answer:

The coordinates of \( B \) are \( (-6,8) \)