Triangle point coordinates

In triangle ABC, determine the coordinates of point B if you know that points A and B lie on the line 3x-y-5=0, points A and C lie on line 2x+3y+4=0, point C lies on the x-coordinate axis, and the angle at vertex C is right.

Final Answer:

Bx =  5.3333
By =  11

Step-by-step explanation:

$$\begin{gathered} A=(x,y) \\ C=(z,0) \\ 3\cdot x-y-5=0 \\ 2\cdot x+3\cdot y+4=0 \\ 3x-y=5 \\ 2x+3y=-4 \\ Row2-\frac{2}{3}\cdot Row1\to Row2 \\ 3.67y=-7.33 \\ y=\frac{-7.33333333}{3.66666667}=-2 \\ x=\frac{5+y}{3}=\frac{5-2}{3}=1 \\ x=1 \\ y=-2 \\ A=(1,-2) \\ B=(x,y) \\ C=(z,0) \\ 2\cdot z+3\cdot 0+4=0 \\ 2z=-4 \\ z=\frac{-4}{2}=-2 \\ z=-2 \\ A=(1,-2) \\ B=(x,y) \\ C=(-2,0) \\ |AB{|}^2=|AC{|}^2+|BC{|}^2 \\ (x-1{)}^2+(y+2{)}^2=(-2-1{)}^2+(0+2{)}^2+(-2-x{)}^2+(0-y{)}^2 \\ -6x+4y-12=0 \\ -6x_1+4y_1-12=0 \\ 3x_1-y_1-5=0 \\ -6\cdot x_1+4\cdot y_1-12=0 \\ 3\cdot x_1-y_1-5=0 \\ 6x_1-4y_1=-12 \\ 3x_1-y_1=5 \\ Row2-\frac{3}{6}\cdot Row1\to Row2 \\ {y}_1=11 \\ {x}_1=\frac{-12+4y_1}{6}=\frac{-12+4\cdot 11}{6}=5.33333333 \\ {x}_1=\frac{16}{3}\doteq 5.333333 \\ Bx=B_x=x_1=5.3333 \end{gathered}$$

$By=B_y=y_1=11$

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