The endpoints of a side of rectangle ABCD in the coordinate plane are at A (3, 7) and B (6, 1). Find the equation of the line that contains the given segment.

Question

The line segment is CD and point C is at (8, 2).
the equation is _____

Answer 1

To find the equation of the line containing the segment AB, we need to determine the slope of the line that passes through points A(3, 7) and B(6, 1).

1. Calculate the slope (m) of line AB:

   \[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{1 - 7}{6 - 3} = \frac{-6}{3} = -2 \]

2. Use the point-slope form of the equation of a line:

   The point-slope form is given by: \[ y - y_1 = m(x - x_1) \]

   We can choose point A(3, 7) for this process:

   \[ y - 7 = -2(x - 3) \]

3. Simplify the equation:

   \[ y - 7 = -2x + 6 \] \[ y = -2x + 6 + 7 \] \[ y = -2x + 13 \]

The equation of the line that contains segment AB (and consequently line CD as per the information given, assuming they are parallel) is:

\[ \boxed{y = -2x + 13} \]

Answer 2

this is incorrect, could there be another answer

Answer 3

Let's re-evaluate what you need. You want to find the equation of the line segment CD, given point C at (8, 2) and the other endpoint potentially needing to be determined.

Step 1: Determine the slope of AB

Given points:

• A(3, 7)
• B(6, 1)

Calculating the slope of line AB: \[ m_{AB} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{1 - 7}{6 - 3} = \frac{-6}{3} = -2 \]

Step 2: Find the coordinates of point D

Since we want to find the coordinates of point D, which will form rectangle ABCD with vertices A, B, and C, the line segment CD should be parallel to the line segment AB.

Since CD is parallel to AB, it will have the same slope of -2.

Step 3: Use point C(8, 2) and the slope to find the equation of line CD

Using the point-slope formula with point C(8, 2) and slope \( m = -2 \):

The point-slope equation is:

\[ y - y_1 = m(x - x_1) \]

Substituting in point C(8, 2):

\[ y - 2 = -2(x - 8) \]

Step 4: Simplifying this equation

\[ y - 2 = -2x + 16 \] \[ y = -2x + 16 + 2 \] \[ y = -2x + 18 \]

Thus, the equation of the line segment CD is:

\[ \boxed{y = -2x + 18} \]

If point D is at the intersection of this line with the vertical line (at some fixed x where D would align to allow rectangle formation), it needs to be calculated based on defined coordinates, but the equation of the line segment is as shown.