Question

select the correct answer from each drop - down menu. point c(3.6, - 0.4) divides $overline{ab}$ in the ratio 3:2. if the coordinates of a are (-6,5), the coordinates of point b are. if point d divides $overline{cb}$ in the ratio 4:5, the coordinates of point d are.

Explanation:

Step1: Find coordinates of B using section - formula for C dividing AB

Let coordinates of B be \((x,y)\). The section - formula for a point \(C(x_c,y_c)\) dividing the line - segment joining \(A(x_a,y_a)\) and \(B(x,y)\) in the ratio \(m:n\) is \(x_c=\frac{mx + nx_a}{m + n}\) and \(y_c=\frac{my+ny_a}{m + n}\). Given \(A(-6,5)\), \(C(3, - 0.4)\) and \(m = 3\), \(n = 2\).
For \(x\) - coordinate: \(3=\frac{3x+2\times(-6)}{3 + 2}\), \(15 = 3x-12\), \(3x=27\), \(x = 9\).
For \(y\) - coordinate: \(-0.4=\frac{3y + 2\times5}{3+2}\), \(-2=3y + 10\), \(3y=-12\), \(y=-4\). So \(B(9,-4)\).

Step2: Find coordinates of D using section - formula for D dividing CB

Let coordinates of D be \((x_d,y_d)\). Now \(C(3,-0.4)\), \(B(9,-4)\) and \(m = 4\), \(n = 5\).
For \(x\) - coordinate: \(x_d=\frac{4\times9+5\times3}{4 + 5}=\frac{36 + 15}{9}=\frac{51}{9}=\frac{17}{3}\).
For \(y\) - coordinate: \(y_d=\frac{4\times(-4)+5\times(-0.4)}{4 + 5}=\frac{-16-2}{9}=\frac{-18}{9}=-2\).

Answer:

The coordinates of point D are \((\frac{17}{3},-2)\)