Answer: [tex]\bold{a)\ y=\dfrac{9}{7}x+\dfrac{18}{7}}[/tex]
b) y = 3x - 6
Step-by-step explanation:
Median is the line from the Vertex to the Midpoint of the opposite side
a)
Step 1: Find the Midpoint of QR:
Q = (4, 6) R = (5, -3)
[tex]x_M=\dfrac{x_Q+x_R}{2}\qquad \qquad \qquad y_M=\dfrac{y_Q+y_R}{2}\\\\\\x_M=\dfrac{4+5}{2}\qquad \qquad \qquad \qquad y_M=\dfrac{6+(-3)}{2}\\\\\\x_M=\dfrac{9}{2} \qquad \qquad \qquad \qquad \qquad y_M=\dfrac{3}{2}[/tex]
[tex]Midpoint_{QR}=\bigg(\dfrac{9}{2},\dfrac{3}{2}\bigg)[/tex]
Step 2: Find the slope (m) for P (-2,0) to Midpoint of QR:
[tex]m=\dfrac{y_2-y_1}{x_2-x_1}\\\\\\m=\dfrac{\frac{9}{2}-0}{\frac{3}{2}-(-2)}\\\\\\m=\dfrac{\frac{9}{2}}{\frac{7}{2}}\\\\\\m=\bigg{\dfrac{9}{7}}[/tex]
Step 3: Find the equation of the line from P to Midpoint of QR:
[tex]y-y_P=m(x-x_P)\\\\\\y-0=\dfrac{9}{7}\bigg(x-(-2)\bigg)\\\\\\\\\large\boxed{y=\dfrac{9}{7}x+\dfrac{18}{7}}[/tex]
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b)
Step 1: Find the Midpoint of PR:
P = (-2, 0) R = (5, -3)
[tex]x_M=\dfrac{x_P+x_R}{2}\qquad \qquad \qquad y_M=\dfrac{y_P+y_R}{2}\\\\\\x_M=\dfrac{-2+5}{2}\qquad \qquad \qquad \qquad y_M=\dfrac{0+(-3)}{2}\\\\\\x_M=\dfrac{3}{2} \qquad \qquad \qquad \qquad \qquad y_M=-\dfrac{3}{2}[/tex]
[tex]Midpoint_{PR}=\bigg(\dfrac{9}{2},\dfrac{3}{2}\bigg)[/tex]
Step 2: Find the slope (m) for Q (4,6) to Midpoint of PR:
[tex]m=\dfrac{y_2-y_1}{x_2-x_1}\\\\\\m=\dfrac{6-(-\frac{3}{2})}{4-(\frac{3}{2})}\\\\\\m=\dfrac{\frac{15}{2}}{\frac{5}{2}}\\\\\\m=\bigg{3}[/tex]
Step 3: Find the equation of the line from Q to Midpoint of PR:
[tex]y-y_Q=m(x-x_Q)\\\\\\y-6=3(x-4)\\\\\\y-6=3x-12\\\\\\y=3x-12+6\\\\\\\large\boxed{y=3x-6}[/tex]
