How to answer this questions: coordinates of the midpoints and finding the equations of lines AC?

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How to answer this questions: coordinates of the midpoints and finding the equations of lines AC?

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Answer 1 · 2018-01-30T14:09:11

$see explanation$ $"see explanation"$

Explanation:

$(a)$ $(a)$

$given A (x_{1}, y_{1}) and B (x_{2}, y_{2})$ $"given "A(x_1,y_1)" and "B(x_2,y_2)$

$then the coordinates of the midpoint (M) of AB$ $"then the coordinates of the midpoint (M) of AB"$

$∙ x M = [\frac{1}{2} (x_{1} + x_{2}), \frac{1}{2} (y_{1} + y_{2})]$ $•color(white)(x)M=[1/2(x_1+x_2),1/2(y_1+y_2)]$

$\Rightarrow M = [\frac{1}{2} (9 + 5), \frac{1}{2} (10 + 0)] = (7, 5)$ $rArrM=[1/2(9+5),1/2(10+0)]=(7,5)$

$\Rightarrow N = [\frac{1}{2} (9 + 13), \frac{1}{2} (10 + 0)] = (11, 5)$ $rArrN=[1/2(9+13),1/2(10+0)]=(11,5)$

$(b)$ $(b)$

$equations of lines AC and BC$ $"equations of lines AC and BC"$

$the equation of a line in slope-intercept form$ $"the equation of a line in "color(blue)"slope-intercept form"$ is.

$∙ x y = m x + b$ $•color(white)(x)y=mx+b$

$where m is the slope and b the y-intercept$ $"where m is the slope and b the y-intercept"$

$calculate the slope m using the gradient formula$ $"calculate the slope m using the "color(blue)"gradient formula"$

$∙ x m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ $•color(white)(x)m=(y_2-y_1)/(x_2-x_1)$

$\Rightarrow m_{A C} = \frac{10 - 0}{9 - 5} = \frac{10}{4} = \frac{5}{2}$ $rArrm_(AC)=(10-0)/(9-5)=10/4=5/2$

$\Rightarrow y = \frac{5}{2} x + b \leftarrow partial equation$ $rArry=5/2x+blarrcolor(blue)"partial equation"$

$to find b substitute the coordinates of either A or C$ $"to find b substitute the coordinates of either A or C"$
$into the partial equation$ $"into the partial equation"$

$using A (5, 0) then$ $"using "A(5,0)" then"$

$0 = \frac{25}{2} + b \Rightarrow b = - \frac{25}{2}$ $0=25/2+brArrb=-25/2$

$\Rightarrow y = \frac{5}{2} x - \frac{25}{2} \leftarrow equation of AC$ $rArry=5/2x-25/2larrcolor(blue)"equation of AC"$

$Similarly for the equation of BC$ $"Similarly for the equation of BC"$

$m_{B C} = \frac{10 - 0}{9 - 13} = \frac{10}{- 4} = - \frac{5}{2}$ $m_(BC)=(10-0)/(9-13)=10/(-4)=-5/2$

$\Rightarrow y = - \frac{5}{2} x + b$ $rArry=-5/2x+b$

$using B (13, 0) then$ $"using "B(13,0)" then"$

$0 = - \frac{65}{2} + b \Rightarrow b = \frac{65}{2}$ $0=-65/2+brArrb=65/2$

$\Rightarrow y = - \frac{5}{2} x + \frac{65}{2} \leftarrow equation of BC$ $rArry=-5/2x+65/2larrcolor(blue)"equation of BC"$

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How to answer this questions: coordinates of the midpoints and finding the equations of lines AC?

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