Answered

Find the best solutions to your questions at Westonci.ca, the premier Q&A platform with a community of knowledgeable experts. Explore in-depth answers to your questions from a knowledgeable community of experts across different fields. Get quick and reliable solutions to your questions from a community of experienced experts on our platform.

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 $C$.

Sagot :

brainsoft

Answer:

D is approximately (-2, -1)

Step-by-step explanation:

[tex]{ \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 \: \: }}}}[/tex]

[tex]{ \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 \: \: }}}}[/tex]

Solve the green equation and blue equation simultaneously:

[tex]{ \boxed{ \tt{ \red{ \: y \approx - 2 \: \: }}and \: \: { \red{x \approx - 1}}}}[/tex]

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

  • Slope of AB =Slope of CD

[tex]\\ \tt\hookrightarrow \dfrac{4-3}{-1-2}=\dfrac{b+2}{a+2}[/tex]

[tex]\\ \tt\hookrightarrow \dfrac{-1}{2}=\dfrac{b+2}{a+2}[/tex]

[tex]\\ \tt\hookrightarrow -a-2=2b+4[/tex]

[tex]\\ \tt\hookrightarrow a+2b+6=0\dots(1)[/tex]

  • Slope of AD=Slope of BC

[tex]\\ \tt\hookrightarrow \dfrac{b-3}{a-2}=\dfrac{-2-4}{-2+1}[/tex]

[tex]\\ \tt\hookrightarrow \dfrac{b-3}{a-2}=6[/tex]

[tex]\\ \tt\hookrightarrow 6a-12=b-3[/tex]

[tex]\\ \tt\hookrightarrow 6a-b-9=0\dots(2)[/tex]

Multiplying 2 with eq(2)

[tex]\\ \tt\hookrightarrow 12a-2b-18=0\dots(3)[/tex]

  • Add eq(1) and (3)

[tex]\\ \tt\hookrightarrow 13a-12=0[/tex]

[tex]\\ \tt\hookrightarrow a=12/13=0.9\to 1[/tex]

  • Put in eq(1)

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

[tex]\\ \tt\hookrightarrow 90/13=-2b[/tex]

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

Visit us again for up-to-date and reliable answers. We're always ready to assist you with your informational needs. We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. Westonci.ca is your go-to source for reliable answers. Return soon for more expert insights.