Question

find the coordinate of point c on ab such that the ratio of ac to cb is 1:4

Explanation:

1. Assume the coordinates of point \(A=(x_1,y_1)\) and point \(B=(x_2,y_2)\). Let's first assume \(A = (- 3,-6)\) and \(B=(6,6)\) (by observing the grid - we can estimate the coordinates of \(A\) and \(B\) from the graph).

• The section - formula for a point \(C=(x,y)\) that divides the line - segment joining \(A=(x_1,y_1)\) and \(B=(x_2,y_2)\) in the ratio \(m:n\) is given by:
• \(x=\frac{mx_2+nx_1}{m + n}\) and \(y=\frac{my_2+ny_1}{m + n}\).
• Here, \(m = 1\) and \(n = 4\), \(x_1=-3\), \(y_1=-6\), \(x_2 = 6\), and \(y_2 = 6\).

1. Calculate the \(x\) - coordinate of point \(C\):

• Substitute the values into the \(x\) - coordinate formula:
• \(x=\frac{1\times6+4\times(-3)}{1 + 4}=\frac{6-12}{5}=\frac{-6}{5}=-1.2\).

1. Calculate the \(y\) - coordinate of point \(C\):

• Substitute the values into the \(y\) - coordinate formula:
• \(y=\frac{1\times6+4\times(-6)}{1 + 4}=\frac{6-24}{5}=\frac{-18}{5}=-3.6\).

Step1: Identify section - formula

Use \(x=\frac{mx_2+nx_1}{m + n}\) and \(y=\frac{my_2+ny_1}{m + n}\) for \(m:n\) division of line - segment.

Step2: Calculate \(x\) - coordinate

\(x=\frac{1\times6 + 4\times(-3)}{1+4}=\frac{6 - 12}{5}=-1.2\).

Step3: Calculate \(y\) - coordinate

\(y=\frac{1\times6+4\times(-6)}{1 + 4}=\frac{6-24}{5}=-3.6\).

Answer:

The coordinates of point \(C\) are \((-1.2,-3.6)\)