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Vadim26 [7]
2 years ago
14

Given points $A(2,3)$, $B(-1,4)$, and $C(-2,-2)$, determine point $D$ so that the slope from $A$ to $B$ equals the slope from $C

$ to $D$, and the slope from $A$ to $D$ equals the slope from $B$ to $C$.
Mathematics
2 answers:
miss Akunina [59]2 years ago
7 0

Answer:

D is approximately (-2, -1)

Step-by-step explanation:

{ \tt{slope \:AB = slope \: CD }} \\  \\ { \tt{ \frac{(4 - 3)}{( - 1 - 2)}  =  \frac{(y - ( - 2))}{(x - ( - 2))} }} \\  \\ { \tt{ \frac{1}{ - 3} =  \frac{y + 2}{x + 2}  }} \\  \\ { \tt{x + 2 =  - 3y - 6}} \\ { \underline{ \tt{ \green{ \:  \: 3y + x =  - 8 \:  \: }}}}

{ \tt{slope \: AD= slope \:  BC}} \\ \\  { \tt{ \frac{y - 3}{x - 2}  =  \frac{ - 2 - 4}{ - 2 - 1} }} \\  \\ { \tt{ \frac{y - 3}{x - 2} =  \frac{6}{3}  }} \\  \\ { \tt{ \frac{y - 3}{x - 2}  = 2}} \\  \\ { \tt{y - 3 = 2(x - 2)}} \\  { \tt{y - 3 = 2x - 4}} \\ { \underline{ \tt{ \blue{ \:  \: y - 2x =  - 1 \:  \: }}}}

Solve the green equation and blue equation simultaneously:

{ \boxed{ \tt{ \red{ \: y \approx - 2 \:  \: }}and \:  \: { \red{x \approx - 1}}}}

gayaneshka [121]2 years ago
4 0

Let the co-ordinates of D be (a,b)

  • Slope of AB =Slope of CD

\\ \tt\hookrightarrow \dfrac{4-3}{-1-2}=\dfrac{b+2}{a+2}

\\ \tt\hookrightarrow \dfrac{-1}{2}=\dfrac{b+2}{a+2}

\\ \tt\hookrightarrow -a-2=2b+4

\\ \tt\hookrightarrow a+2b+6=0\dots(1)

  • Slope of AD=Slope of BC

\\ \tt\hookrightarrow \dfrac{b-3}{a-2}=\dfrac{-2-4}{-2+1}

\\ \tt\hookrightarrow \dfrac{b-3}{a-2}=6

\\ \tt\hookrightarrow 6a-12=b-3

\\ \tt\hookrightarrow 6a-b-9=0\dots(2)

Multiplying 2 with eq(2)

\\ \tt\hookrightarrow 12a-2b-18=0\dots(3)

  • Add eq(1) and (3)

\\ \tt\hookrightarrow 13a-12=0

\\ \tt\hookrightarrow a=12/13=0.9\to 1

  • Put in eq(1)

\\ \tt\hookrightarrow 12/13+2b+6=0

\\ \tt\hookrightarrow 90/13=-2b

\\ \tt\hookrightarrow b=-90/26=-3 4\to 3

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