Question

question
follow the guided instructions below to rotate the figure $90^\circ$ counter-clockwise about the point $(3, -1)$.

now draw a line through the yellow center that is perpendicular to
the line that you just drew.

Explanation:

Step1: Identify the slope of the original line

First, we need to find the slope of the dashed line (the line we just drew for rotation). Let's assume the dashed line passes through two points, say \((3, -1)\) (the center of rotation) and another point, but from the graph, the dashed line seems to have a slope of \(1\) (since it goes through points like \((-3, -7)\) and \((6, 2)\), the rise over run is \(\frac{2 - (-7)}{6 - (-3)}=\frac{9}{9} = 1\)).

Step2: Determine the slope of the perpendicular line

The slope of a line perpendicular to a line with slope \(m\) is \(-\frac{1}{m}\). Since the original line has a slope of \(1\), the slope of the perpendicular line is \(-\frac{1}{1}=-1\).

Step3: Draw the line through the yellow center with slope -1

The yellow center is at \((3, -1)\). Using the slope - 1, we can draw a line that goes through \((3, -1)\) and has a slope of - 1. For example, from \((3, -1)\), moving 1 unit to the right (increase \(x\) by 1) and 1 unit down (decrease \(y\) by 1) gives the point \((4, -2)\), and moving 1 unit to the left (decrease \(x\) by 1) and 1 unit up (increase \(y\) by 1) gives the point \((2, 0)\). We can draw a line connecting these points (or using the slope - 1) through the yellow center \((3, -1)\).

Answer:

The line through the yellow center \((3, - 1)\) with a slope of \(-1\) (perpendicular to the original line with slope \(1\)) is drawn. (In a graphical context, this line will have a negative slope and pass through \((3, -1)\), for example, passing through points like \((2, 0)\) and \((4, -2)\) or similar points with a slope of \(-1\) through \((3, -1)\))