Question

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 $C$.

Answer 1

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

• Slope of AB =Slope of CD

`\\ \tt\hookrightarrow (4-3)/(-1-2)=(b+2)/(a+2)`

`\\ \tt\hookrightarrow (-1)/(2)=(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 (b-3)/(a-2)=(-2-4)/(-2+1)`

`\\ \tt\hookrightarrow (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`

Answer 2

D is approximately (-2, -1)

Explanation:

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

`{ \tt{slope \: AD= slope \: BC}} \\ \\ { \tt{ (y - 3)/(x - 2) = ( - 2 - 4)/( - 2 - 1) }} \\ \\ { \tt{ (y - 3)/(x - 2) = (6)/(3) }} \\ \\ { \tt{ (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}}}}`