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]
