To find the equation of the median [tex]\( \overline{AD} \)[/tex] in standard form [tex]\( Ax + By = C \)[/tex], let's follow the steps step-by-step.
### Step 1: Find the Midpoint [tex]\( D \)[/tex] of [tex]\( \overline{BC} \)[/tex]
The midpoint [tex]\( D \)[/tex] of a line segment with endpoints [tex]\( B(x_1, y_1) \)[/tex] and [tex]\( C(x_2, y_2) \)[/tex] is calculated as:
[tex]\[ D \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \][/tex]
Substitute the coordinates of [tex]\( B \)[/tex] and [tex]\( C \)[/tex]:
[tex]\[ B = (-9, 8), \quad C = (-7, 6) \][/tex]
Calculate the midpoint:
[tex]\[ D_x = \frac{-9 + (-7)}{2} = \frac{-16}{2} = -8 \][/tex]
[tex]\[ D_y = \frac{8 + 6}{2} = \frac{14}{2} = 7 \][/tex]
Thus, the coordinates of [tex]\( D \)[/tex] are [tex]\( D(-8, 7) \)[/tex].
### Step 2: Find the Slope [tex]\( m \)[/tex] of [tex]\( \overline{AD} \)[/tex]
The slope [tex]\( m \)[/tex] of a line passing through points [tex]\( A(x_1, y_1) \)[/tex] and [tex]\( D(x_2, y_2) \)[/tex] is calculated as:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Substitute the coordinates of [tex]\( A \)[/tex] and [tex]\( D \)[/tex]:
[tex]\[ A = (-10, 5), \quad D = (-8, 7) \][/tex]
Calculate the slope:
[tex]\[ m = \frac{7 - 5}{-8 - (-10)} = \frac{2}{2} = 1 \][/tex]
### Step 3: Find the Equation of the Line in Slope-Intercept Form
The equation of a line in slope-intercept form is:
[tex]\[ y = mx + b \][/tex]
Substitute the slope [tex]\( m = 1 \)[/tex]:
[tex]\[ y = x + b \][/tex]
To find [tex]\( b \)[/tex], substitute the coordinates of point [tex]\( A(-10, 5) \)[/tex]:
[tex]\[ 5 = -10 + b \][/tex]
[tex]\[ b = 5 + 10 \][/tex]
[tex]\[ b = 15 \][/tex]
Thus, the equation of the line in slope-intercept form is:
[tex]\[ y = x + 15 \][/tex]
### Step 4: Convert to Standard Form [tex]\( Ax + By = C \)[/tex]
Rearrange the equation [tex]\( y = x + 15 \)[/tex] into standard form:
[tex]\[ y - x = 15 \][/tex]
Rewrite it to match the standard form [tex]\( Ax + By = C \)[/tex]:
[tex]\[ -x + y = 15 \][/tex]
Thus, the coefficients are:
[tex]\[ A = -1, \quad B = 1, \quad C = 15 \][/tex]
Therefore, the equation of the median [tex]\( \overline{AD} \)[/tex] in standard form is:
[tex]\[ -x + y = 15 \][/tex]
and the values are:
[tex]\[ A = -1, \quad B = 1, \quad C = 15 \][/tex]
