Question

The vertices of a quadrilateral are A(-1, 1), B(2, 4), C(2, -4), and D(0, -2).

a. Write an equation for the line through A and B.
b. Write an equation for the line through C and D.
c. Without graphing the lines, what can you tell about the lines from their slopes?
a) Both lines have positive slopes.
b) Both lines have negative slopes.
c) The lines have perpendicular slopes.
d) The lines have parallel slopes.

Answer 1

The equation for the line through points A and B is y = x + 2, and for the line through points C and D is y = -x + 2. These lines have slopes that are opposite reciprocals of each other, indicating that they are perpendicular to each other.

Step-by-step explanation:

Finding Equations of Lines and Analyzing Slopes

To write an equation for the line through points A(-1, 1) and B(2, 4), we first find the slope (m) which is change in y over change in x. The slope of AB is calculated as (4 - 1) /(2 - (-1)) = 3/3 = 1. So, the line has a positive slope. Using point A to write the equation in point-slope form gives us y - 1 = 1(x + 1), which simplifies to y = x + 2.

For the line through points C(2, -4) and D(0, -2), the slope is (-2 - (-4))/(0 - 2) = 2/-2 = -1. This line also has a consistent slope but it's negative. The point-slope form using point C is y + 4 = -1(x - 2), leading to y = -x + 2.

Comparing the slopes, line AB has a positive slope while line CD has a negative slope, so option a) is incorrect and b) is also incorrect. Because the slopes are opposite reciprocals (1 and -1), they are perpendicular slopes; therefore, option c) is correct and d) is incorrect.