AA, BBB, and CCC are collinear, and BBB is between AAA and CCC. The ratio of ABABA, B to ACACA, C is 3:43:43, colon, 4. If AAA is at (-8,1)(−8,1)(, minus, 8, comma, 1, )and BBB is at (-2,-2)(−2,−2)(, minus, 2, comma, minus, 2, ), what are the coordinates of point CCC?

Accepted Solution

Answer:The coordinates of point C are (0,-3).Step-by-step explanation:It is given that A, B, and C are collinear and B is between A and C.The ratio of AB to AC is 3:4. Let length of AB and AC be 3x and 4x respectively.[tex]AC=AB+BC[/tex][tex]4x=3x+BC[/tex][tex]x=BC[/tex][tex]\frac{AB}{BC}=\frac{3x}{x}=3:1[/tex].Therefore, AB to BC is 3:1.The given ordered pairs are A(-8,1) and B(-2,-2).Let as assume that the coordinate of C is (a,b).Section formula: If a point divides a line segment in m:n whose end points are [tex](x_1,y_1)[/tex] and [tex](x_2,y_2)[/tex], then the coordinates of that point are [tex](\frac{mx_2+nx_1}{m+n},\frac{my_2+ny_1}{m+n})[/tex]Point B divided the line AC is 3:1. Using section formula we get[tex]B=(\frac{(3)(a)+(1)(-8)}{3+1},\frac{(3)(b)+(1)(1)}{(3)+(1)})[/tex][tex]B=(\frac{3a-8}{4},\frac{3b+1}{4})[/tex]The coordinates of B are (-2,-2).[tex](-2,-2)=(\frac{3a-8}{4},\frac{3b+1}{4})[/tex]On comparing both sides.[tex]-2=\frac{3a-8}{4}[/tex][tex]-8=3a-8[/tex][tex]0=3a[/tex][tex]a=0[/tex]The value of a is 0.[tex]-2=\frac{3b+1}{4}[/tex][tex]-8=3b+1[/tex][tex]-9=3b[/tex][tex]-3=b[/tex]The value of b is -3.Therefore the coordinates of point C are (0,-3).