Question

use tools select the diagram that shows the following: points a(4, 3), b(2, 1), c, and d(-2, -3) are collinear. points e(-3, 1) and f are collinear on a line that intersects $overleftrightarrow{ad}$ at point b.

Explanation:

Step1: Recall slope - formula

The slope formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $m=\frac{y_2 - y_1}{x_2 - x_1}$.

Step2: Calculate slope of line passing through \(A(4,3)\) and \(B(2,1)\)

$m_{AB}=\frac{1 - 3}{2 - 4}=\frac{-2}{-2}=1$.

Step3: Calculate slope of line passing through \(B(2,1)\) and \(D(-2,-3)\)

$m_{BD}=\frac{-3 - 1}{-2 - 2}=\frac{-4}{-4}=1$.
Since $m_{AB}=m_{BD}=1$, points \(A\), \(B\) and \(D\) are collinear.

Step4: Check the position of \(E(-3,1)\) and \(F\) with respect to line \(AD\) intersecting at \(B\)

We need to check if we can draw a line through \(E(-3,1)\) and \(F\) that intersects the line \(AD\) at \(B(2,1)\).
For option A, we can visually see that the points \(A(4,3)\), \(B(2,1)\), \(D(-2,-3)\) are on the same straight - line (as the slope calculation verified) and we can draw a line through \(E(-3,1)\) and \(F\) that intersects the line \(AD\) at \(B\).

Answer:

A