Final answer:
Lines A and D are parallel, lines B and D are perpendicular, and lines A and C are perpendicular. Lines A and B are not perpendicular and lines B and C are not parallel.
Step-by-step explanation:
Determining Relationships Between Lines
To determine the relationships between the lines A, B, C, and D, we need to find their slopes and compare them. In the form y = mx + b, the coefficient of x represents the slope (m). Lines that are perpendicular to each other have slopes that are negative reciprocals, while parallel lines have equal slopes.
- Equation A in slope-intercept form is y = (4/3)x + (5/3), so its slope is 4/3.
- Equation B is y = (3/4)x - (1/4), with a slope of 3/4.
- Equation C can be rewritten as y = (-3/4)x + (7/4), giving it a slope of -3/4.
- Equation D is y = (-4/3)x + (2/3), so its slope is -4/3.
By examining these slopes:
- Lines A and B are not perpendicular since the slope of A (4/3) is not the negative reciprocal of B's slope (3/4).
- Lines A and D are parallel because they have the same absolute slope value (4/3), but opposite signs.
- Lines B and D are perpendicular because the slope of B (3/4) is the negative reciprocal of D's slope (-4/3).
- Lines B and C are not parallel or perpendicular; their slopes are not equal nor negative reciprocals of each other.
- Lines A and C are perpendicular because the slope of A (4/3) is the negative reciprocal of C's slope (-3/4).