Question

in the diagram, point d divides line segment ab in the ratio of 5:3. if line segment ac is vertical and line segment cd is horizontal, what are the coordinates of point c?
a=(2, - 6)
b=(10,2)
c=(?)
a. (7, - 1)
b. (5, - 3)
c. (2, - 3)
d. (2, - 1)

Explanation:

Step1: Find the coordinates of point D using section - formula

The section formula for a point \(D(x,y)\) that divides the line - segment joining \(A(x_1,y_1)\) and \(B(x_2,y_2)\) in the ratio \(m:n\) is \(x=\frac{mx_2+nx_1}{m + n}\) and \(y=\frac{my_2+ny_1}{m + n}\). Here, \(A(2,-6)\), \(B(10,2)\), \(m = 5\), and \(n = 3\).

$$x_D=\frac{5\times10+3\times2}{5 + 3}=\frac{50 + 6}{8}=\frac{56}{8}=7$$

$$y_D=\frac{5\times2+3\times(-6)}{5 + 3}=\frac{10-18}{8}=\frac{-8}{8}=-1$$

So, the coordinates of point \(D\) are \((7,-1)\).

Step2: Determine the coordinates of point C

Since \(AC\) is vertical and \(CD\) is horizontal, the \(x\) - coordinate of \(C\) is the same as the \(x\) - coordinate of \(A\) (because \(AC\) is vertical), and the \(y\) - coordinate of \(C\) is the same as the \(y\) - coordinate of \(D\) (because \(CD\) is horizontal). The \(x\) - coordinate of \(A\) is \(2\) and the \(y\) - coordinate of \(D\) is \(-1\). So, the coordinates of \(C\) are \((2,-1)\).

Answer:

D. \((2,-1)\)