Question

find the coordinates of point c on $overline{ab}$ such that the ratio of ac to cb is 3:6. point c is located at __________.

Explanation:

Step1: Assume coordinates of A and B

Let \(A=(x_1,y_1)\) and \(B=(x_2,y_2)\). From the graph, assume \(A = (- 3,-6)\) and \(B=(5,6)\). The ratio \(AC:CB = 3:6=1:2\), so \(m = 1\) and \(n = 2\).

Step2: Use the section - formula for x - coordinate

The formula for the x - coordinate of the point \(C\) that divides the line segment joining \((x_1,y_1)\) and \((x_2,y_2)\) in the ratio \(m:n\) is \(x=\frac{mx_2+nx_1}{m + n}\).
Substitute \(x_1=-3\), \(x_2 = 5\), \(m = 1\), and \(n = 2\) into the formula:
\[x=\frac{1\times5+2\times(-3)}{1 + 2}=\frac{5-6}{3}=-\frac{1}{3}\]

Step3: Use the section - formula for y - coordinate

The formula for the y - coordinate of the point \(C\) that divides the line segment joining \((x_1,y_1)\) and \((x_2,y_2)\) in the ratio \(m:n\) is \(y=\frac{my_2+ny_1}{m + n}\).
Substitute \(y_1=-6\), \(y_2 = 6\), \(m = 1\), and \(n = 2\) into the formula:
\[y=\frac{1\times6+2\times(-6)}{1 + 2}=\frac{6 - 12}{3}=-2\]

Answer:

\((-\frac{1}{3},-2)\)