Question

reasoning

1. $overline{ab}$ passes through $a(-2,3)$ and $b(4,6)$.

(a) plot $a$ and $b$ and draw $overline{ab}$.
(b) point $c$, at $c(4,2)$, does not lie on $overline{ab}$. plot and label $c$. according to the parallel line postulate, how many lines can be drawn through $c$ that are parallel to $overline{ab}$? one line
(c) draw all the lines that pass through $c$ and are parallel to $overline{ab}$. (hint: think about the slope of $overline{ab}$.)

Explanation:

Step1: Plot points A and B

On a coordinate - grid, mark point A at (-2,3) and point B at (4,6). Then draw a line segment $\overline{AB}$.

Step2: Apply parallel line postulate

The Parallel Line Postulate states that through a point not on a given line, there is exactly one line parallel to the given line. Since point C(4,2) is not on $\overline{AB}$, there is one line through C parallel to $\overline{AB}$.

Step3: Find slope of $\overline{AB}$ and draw parallel line

The slope of $\overline{AB}$ using the formula $m=\frac{y_2 - y_1}{x_2 - x_1}$ where $(x_1,y_1)=(-2,3)$ and $(x_2,y_2)=(4,6)$ is $m=\frac{6 - 3}{4-(-2)}=\frac{3}{6}=\frac{1}{2}$. A line parallel to $\overline{AB}$ through C(4,2) will have the same slope $\frac{1}{2}$. Using the point - slope form $y - y_1=m(x - x_1)$ with $(x_1,y_1)=(4,2)$ and $m = \frac{1}{2}$, we can find points on the line and draw it.

Answer:

(a) Points A and B are plotted on the coordinate - grid and $\overline{AB}$ is drawn.
(b) One line.
(c) A line with slope $\frac{1}{2}$ passing through C(4,2) is drawn.