For Parallel and Perpendicular Lines: Mastery Test

Type the correct answer in each box. If necessary, use / for the fraction bar(s).

Given:
[tex]\[ AB \parallel CD \][/tex]

If the coordinates of point A are [tex]\((8, 0)\)[/tex] and the coordinates of point B are [tex]\((3, 7)\)[/tex], the equation of line [tex]\(AB\)[/tex] is [tex]\(y = \frac{7}{5}x - \frac{56}{5}\)[/tex].

If the coordinates of point D are [tex]\((5, 5)\)[/tex], the equation of line [tex]\(CD\)[/tex] is [tex]\(y = \frac{7}{5}x - \frac{14}{5}\)[/tex].



Answer :

To determine the equations of the lines AB and CD, let's follow a step-by-step solution.

Step 1: Find the slope of line AB

The coordinates of points A and B are given as:
[tex]\[ A = (8, 0) \][/tex]
[tex]\[ B = (3, 7) \][/tex]

The formula for the slope (m) of a line passing through two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]

Using the coordinates of A and B:
[tex]\[ m_{AB} = \frac{7 - 0}{3 - 8} = \frac{7}{-5} = -\frac{7}{5} \][/tex]

So, the slope of line AB is [tex]\(-\frac{7}{5}\)[/tex].

Step 2: Determine the y-intercept of line AB

The equation of a line in slope-intercept form is given by:
[tex]\[ y = mx + b \][/tex]
where [tex]\(m\)[/tex] is the slope and [tex]\(b\)[/tex] is the y-intercept.

We can use point A [tex]\((8, 0)\)[/tex] to find the y-intercept [tex]\(b\)[/tex]:
[tex]\[ 0 = -\frac{7}{5} \cdot 8 + b \][/tex]
[tex]\[ 0 = -\frac{56}{5} + b \][/tex]
[tex]\[ b = \frac{56}{5} \][/tex]

So, the y-intercept of line AB is [tex]\(\frac{56}{5}\)[/tex].

Therefore, the equation of line AB is:
[tex]\[ y = -\frac{7}{5}x + \frac{56}{5} \][/tex]

Step 3: Find the slope of line CD

Since lines AB and CD are parallel, they have the same slope. Thus, the slope of line CD is also [tex]\(-\frac{7}{5}\)[/tex].

Step 4: Determine the y-intercept of line CD

The coordinates of point C are given as [tex]\((5, 5)\)[/tex]. Using it to find the y-intercept of line CD:
[tex]\[ y = -\frac{7}{5}x + b \][/tex]

Substituting the point [tex]\((5, 5)\)[/tex]:
[tex]\[ 5 = -\frac{7}{5} \cdot 5 + b \][/tex]
[tex]\[ 5 = -7 + b \][/tex]
[tex]\[ b = 12 \][/tex]

So, the y-intercept of line CD is 12.

Therefore, the equation of line CD is:
[tex]\[ y = -\frac{7}{5}x + 12 \][/tex]