Question

1. find point d on $overline{ab}$ that is $\frac{3}{4}$ of the distance from a to b.

Explanation:

Step1: Assume coordinates of A and B

Let \(A=(x_1,y_1)\) and \(B=(x_2,y_2)\). From the graph, assume \(A = (- 4,4)\) and \(B=(4, - 2)\).

Step2: Use the section - formula

The formula for the 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, \(m = 3\) and \(n = 1\) (since \(D\) is \(\frac{3}{4}\) of the distance from \(A\) to \(B\), so the ratio of \(AD\) to \(DB\) is \(3:1\)).
For the \(x\) - coordinate of \(D\):
\[x=\frac{3\times4+1\times(-4)}{3 + 1}=\frac{12-4}{4}=\frac{8}{4}=2\]
For the \(y\) - coordinate of \(D\):
\[y=\frac{3\times(-2)+1\times4}{3 + 1}=\frac{-6 + 4}{4}=\frac{-2}{4}=-\frac{1}{2}\]

Answer:

\(D=(2,-\frac{1}{2})\)