Question

1. draw both a line that is parallel and a line that is perpendicular to the given line and goes through the given point.

Explanation:

Step1: Find the slope of the given line

Count the rise - over - run. If we take two points on the given line, say two consecutive intersection points with the grid lines. The line is decreasing, and if we move 1 unit to the right, it moves down 2 units. So the slope $m_1=- 2$.

Step2: Find the slope of the parallel line

Parallel lines have the same slope. So the slope of the parallel line $m_{parallel}=-2$.

Step3: Find the slope of the perpendicular line

The product of the slopes of two perpendicular lines is - 1. Let the slope of the perpendicular line be $m_{perpendicular}$. Then $m_1\times m_{perpendicular}=-1$. Since $m_1 = - 2$, we have $-2\times m_{perpendicular}=-1$, so $m_{perpendicular}=\frac{1}{2}$.

Step4: Use the point - slope form to draw the lines

The point - slope form of a line is $y - y_1=m(x - x_1)$, where $(x_1,y_1)$ is the given point. To draw the lines, start at the given point. For the parallel line, from the given point, move 1 unit to the right and 2 units down repeatedly to plot more points and then draw the line. For the perpendicular line, from the given point, move 2 units to the right and 1 unit up repeatedly to plot more points and then draw the line.

This is a visual - drawing problem, but the key steps involve slope - related calculations. In a coordinate - grid, we can't give an exact numerical answer for the drawn lines in this format. The main idea is to use the slope properties of parallel and perpendicular lines to draw them through the given point.

Answer:

To draw the parallel line: Start at the given point, use a slope of - 2 (move 1 unit right and 2 units down to plot additional points and then draw the line). To draw the perpendicular line: Start at the given point, use a slope of $\frac{1}{2}$ (move 2 units right and 1 unit up to plot additional points and then draw the line).