Question

1. what are the coordinates of the point \\( \frac { 1 } { 4 } \\) of the way from \\( a ( - 6, - 3 ) \\) to \\( b ( 10,9 ) \\)?

Explanation:

Step1: Find the difference in \(x\)-coordinates and \(y\)-coordinates

The formula for the difference in \(x\)-coordinates \(\Delta x=x_2 - x_1\), and for \(y\)-coordinates \(\Delta y=y_2 - y_1\).
Given \(A(-6,-3)\) and \(B(10,9)\), \(\Delta x=10-(-6)=16\), \(\Delta y=9-(-3)=12\).

Step2: Calculate the \(x\)-coordinate of the point

The \(x\)-coordinate of the point \(P\) which is \(\frac{1}{4}\) of the way from \(A\) to \(B\) is \(x=x_1+\frac{1}{4}\Delta x\).
Substitute \(x_1=-6\) and \(\Delta x = 16\) into the formula: \(x=-6+\frac{1}{4}\times16=-6 + 4=-2\).

Step3: Calculate the \(y\)-coordinate of the point

The \(y\)-coordinate of the point \(P\) which is \(\frac{1}{4}\) of the way from \(A\) to \(B\) is \(y=y_1+\frac{1}{4}\Delta y\).
Substitute \(y_1=-3\) and \(\Delta y = 12\) into the formula: \(y=-3+\frac{1}{4}\times12=-3 + 3=0\).

Answer:

The coordinates of the point are \((-2,0)\)