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

$\dfrac{\overrightarrow{AC}}{\overrightarrow{CB}}=\dfrac{3}{1},$

where

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

Then

$\dfrac{a+5}{7-a}=\dfrac{3}{1}\text{ and }\dfrac{b-4}{-2-b}=\dfrac{3}{1}.$

Solve these two equations:

1.

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

2.

$b-4=3(-2-b),\\ \\b-4=-6-3b,\\ \\b+3b=-6+4,\\ \\4b=-2,\\ \\b=2\dfrac{2}{4}=-\dfrac{1}{2}=-0.5.$

Answer: C(4,-0.5)