SAT-Math-12: Perpendicular Lines and Negative Reciprocal Slopes
This lesson covers the geometric and algebraic relationship between perpendicular lines, teaching students how to find and apply negative reciprocal slopes to solve coordinate geometry problems.
1. The Perpendicular Rule
While parallel lines run in the same direction, perpendicular lines cross each other at a perfect $90^\circ$ right angle.
Because they meet at a right angle, their slopes have an inverted relationship. The slopes of perpendicular lines are negative reciprocals (teskari va qarama-qarshi ishorali sonlar) of each other.
$$\text{If Line 1 has slope } m_1, \text{ then the perpendicular Line 2 has slope } m_2 = -\frac{1}{m_1}$$
Another way to test this: if you multiply the two slopes together, the result must always be $-1$.
$$m_1 \cdot m_2 = -1$$
How to Find a Negative Reciprocal (Tezkor usul)
Turn the slope into a fraction.
Flip the fraction upside down.
Change the sign (positive to negative, or negative to positive).
Original Slope (m1) | Perpendicular Slope (m2) |
|---|---|
$\frac{2}{3}$ | $-\frac{3}{2}$ |
$-4$ (which is $-\frac{4}{1}$) | $+\frac{1}{4}$ |
$1$ | $-1$ |
🇺🇿 Uzbek Explanation:
Perpendikulyar chiziqlar: Bir-biri bilan $90^\circ$ li tik burchak ostida kesishadigan chiziqlar perpendikulyar chiziqlar deyiladi. Ularning nishabliklari (slope) bir-biriga teskari va qarama-qarshi bo'ladi. Masalan, birinchi chiziqning nishabligi $\frac{3}{5}$ bo'lsa, unga perpendikulyar bo'lgan chiziqniki $-\frac{5}{3}$ bo'ladi (kasrni teskari qilib, ishorasini o'zgartirasiz).
2. High-Yield SAT Practice Question
Question: Line $j$ passes through the points $(2, 5)$ and $(4, 9)$. Line $k$ is perpendicular to line $j$ in the $xy$-coordinate plane. What is the slope of line $k$?
A) $2$
B) $\frac{1}{2}$
C) $-2$
D) $-\frac{1}{2}$
Step-by-Step Explanation:
Find the slope of Line $j$ using our two-point slope formula ($m = \frac{y_2 - y_1}{x_2 - x_1}$):
$$m_j = \frac{9 - 5}{4 - 2} = \frac{4}{2} = 2$$
Apply the perpendicular rule: The slope of Line $j$ is $2$ (or $\frac{2}{1}$).
Flip and switch the sign: * Flip $\frac{2}{1} \rightarrow \frac{1}{2}$
Change positive to negative $\rightarrow -\frac{1}{2}$
Therefore, the slope of the perpendicular line $k$ must be $-\frac{1}{2}$.
Correct Answer: D
Summary
Perpendicular = Negative Reciprocal Slopes: Flip the fraction and flip the sign.
The algebraic test for perpendicular slopes is always $m_1 \cdot m_2 = -1$.
Horizontal lines ($m=0$) and vertical lines ($m=\text{undefined}$) are always perpendicular to each other.
Time for a well-deserved rest, my bro! ☕ You are doing an awesome job structuring this curriculum. Whenever you are fully recharged and ready to jump back in, just drop a message and we will kick off with SAT-Math-13: Solving Multi-Step Linear Inequalities!