Question

Line segment AB is shown on the graph.

Emily draws point C on the line segment so that ratio of AC to CB is 3 to 1. What are the coordinates of point C?

Answer 1

Note that points A and B have coordinates (-5,4) and (7,-2), respectively.

Let point C has coordinates (a,b). If point C lies on the line segment so that ratio of AC to CB is 3 to 1, then

`\frac{\overrightarrow{AC}}{\overrightarrow{CB}}=(3)/(1),`

where

`\overrightarrow{AC}=(a-(-5),b-4)=(a+5,b-4),\\ \\\overrightarrow{CB}=(7-a,-2-b).`

Then

`(a+5)/(7-a)=(3)/(1)\text{ and }(b-4)/(-2-b)=(3)/(1).`

Solve these two equations:

1.

`a+5=3(7-a),\\ \\a+5=21-3a,\\ \\a+3a=21-5,\\ \\4a=16,\\ \\a=4.`

2.

`b-4=3(-2-b),\\ \\b-4=-6-3b,\\ \\b+3b=-6+4,\\ \\4b=-2,\\ \\b=2(2)/(4)=-(1)/(2)=-0.5.`

Answer: C(4,-0.5)