Parallel and Perpendicular Lines on the SAT
Slope Relationships
Build equations from context, spot patterns fast, and practice with intent.
Why the SAT Emphasizes Parallel and Perpendicular Lines
Parallel and perpendicular lines show up in coordinate geometry questions that look simple but move quickly. The SAT wants you to recognize slope relationships without spending too much time graphing.
In this lesson, you will learn a reliable way to find the slope of a line parallel or perpendicular to a given line and then build the equation through a point. You will also see special cases like vertical and horizontal lines that often trip students up.
A Simple Definition Unlocks Parallel and Perpendicular Lines
Parallel lines have the same slope, so if one line has slope $ \frac{3}{4}$, any parallel line also has slope $ \frac{3}{4}$. Perpendicular lines have slopes that multiply to $-1$.
That perpendicular rule means you flip the fraction and change the sign. If the slope is $ \frac{3}{4}$, the perpendicular slope is $- \frac{4}{3}$. If the original line is vertical or horizontal, the perpendicular line is the opposite type.
Work Through Parallel and Perpendicular Lines Step by Step
A line has slope $ \frac{3}{4}$. What is the slope of a perpendicular line?
Find the given line's slope first, then apply the parallel or perpendicular rule before writing the new equation.
Start with the given slope $\frac{3}{4}$.
Flip the fraction as the first step toward a negative reciprocal.
Change the sign to make the slope perpendicular.
Use Desmos to Check Parallel and Perpendicular Lines
A line has slope $ \frac{3}{4}$. What is the slope of a perpendicular line?
Desmos is useful when you want to verify perpendicularity visually. Graph both lines and check that they intersect at a right angle.
y = (3/4)x + 2
y = (-4/3)x + 1
Algebra is faster for slope calculations, but Desmos is a strong check when slopes are messy or when the graph is part of the answer choices.
Desmos is faster when you want to check multiple answer choices visually. Algebra is faster when the slope relationship is simple and you can compute the negative reciprocal quickly.
Expert move: Graph the lines and use intersections for "when do they match?" questions; Desmos also verifies slope and intercepts quickly when coefficients are messy.
When to skip Desmos: For quick slope or parallel/perpendicular checks, algebra is faster; use Desmos for verification.
- Desmos features used: graphing, slope comparison.
- Common mistake: using the reciprocal but forgetting the negative sign.
Practice Parallel and Perpendicular Lines with SAT-Style Questions
Focus on slope relationships rather than drawing full graphs.
Which slope is parallel to the line ?
A line has slope . What is the slope of a perpendicular line?
Line passes through and . Which line is parallel to ?
Which equation is perpendicular to ?
Key Takeaways to Remember for Parallel and Perpendicular Lines
- Parallel lines have equal slopes.
- Perpendicular slopes are negative reciprocals.
- Desmos helps verify angle relationships when graphs are involved.
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