Question

5 on the coordinate plane, draw:
a. a line m that is a
translation of line ě.
b. a line n that is a
rotation of line ě, using
the origin as the center
of rotation.

Explanation:

Step1: Understanding translation

A translation of a line means sliding the line without rotating or resizing it. So, line \(m\) will have the same slope as line \(\ell\). For example, if line \(\ell\) has a slope \(m = \frac{y_2 - y_1}{x_2 - x_1}\) (say passing through \((0,0)\) and \((4,4)\) so slope \(m = 1\)), then line \(m\) can be drawn by moving every point of line \(\ell\) the same distance in the same direction. For instance, if we translate line \(\ell\) 2 units up, then a point \((x,y)\) on \(\ell\) moves to \((x,y + 2)\).

Step2: Understanding rotation

A rotation of a line about the origin. If we rotate line \(\ell\) (with slope \(1\)) by \(90^{\circ}\) counter - clockwise about the origin. The slope of a line with slope \(m\) after a \(90^{\circ}\) counter - clockwise rotation about the origin is \(m'=-\frac{1}{m}\). So if \(m = 1\), the slope of the rotated line \(n\) (after \(90^{\circ}\) counter - clockwise rotation) is \(- 1\). We can use the transformation rule for rotation: if a point \((x,y)\) is rotated \(90^{\circ}\) counter - clockwise about the origin, it becomes \((-y,x)\). For a line \(\ell\) passing through \((0,0)\) and \((4,4)\), after \(90^{\circ}\) counter - clockwise rotation, the points \((0,0)\) remains \((0,0)\) and \((4,4)\) becomes \((-4,4)\)

Answer:

a. To draw line \(m\) (translation of \(\ell\)):

• If \(\ell\) has a slope of \(1\) (assuming it passes through \((0,0)\) and \((4,4)\)), choose a translation vector (e.g., \((2,0)\)). Then if a point \((x,y)\) is on \(\ell\), the corresponding point \((x + 2,y)\) is on \(m\). So if \(\ell\) has points \((0,0)\) and \((4,4)\), \(m\) will have points \((2,0)\) and \((6,4)\) and we can draw the line passing through these new points.

b. To draw line \(n\) (rotation of \(\ell\) about the origin):

• If we rotate \(\ell\) (with slope \(1\)) \(90^{\circ}\) counter - clockwise about the origin. Using the rotation rule \((x,y)\to(-y,x)\). If \(\ell\) passes through \((0,0)\) and \((4,4)\), after rotation, the line \(n\) passes through \((0,0)\) and \((-4,4)\). Draw the line passing through \((0,0)\) and \((-4,4)\) (slope \(m=-1\))