import React, { useRef, useState, useEffect } from "react"; import { ArrowDown, Check, BookOpen, Layers } from "lucide-react"; import ParallelPerpendicularWidget from "../../../components/lessons/ParallelPerpendicularWidget"; import Quiz from "../../../components/lessons/Quiz"; import { LINEAR_PARALLEL_PERP_QUIZ_DATA } from "../../../utils/constants"; import { Frac } from "../../../components/Math"; interface LessonProps { onFinish?: () => void; } const LinearParallelPerpendicularLesson: React.FC = ({ onFinish, }) => { const [activeSection, setActiveSection] = useState(0); const sectionsRef = useRef<(HTMLElement | null)[]>([]); const scrollToSection = (index: number) => { setActiveSection(index); sectionsRef.current[index]?.scrollIntoView({ behavior: "smooth", block: "start", }); }; useEffect(() => { const observer = new IntersectionObserver( (entries) => { entries.forEach((entry) => { if (entry.isIntersecting) { const index = sectionsRef.current.indexOf( entry.target as HTMLElement, ); if (index !== -1) setActiveSection(index); } }); }, { rootMargin: "-20% 0px -60% 0px" }, ); sectionsRef.current.forEach((section) => { if (section) observer.observe(section); }); return () => observer.disconnect(); }, []); const SectionMarker = ({ index, title, icon: Icon, }: { index: number; title: string; icon: any; }) => { const isActive = activeSection === index; const isPast = activeSection > index; return ( ); }; return (
{/* Section 1 */}
{ sectionsRef.current[0] = el; }} className="min-h-screen flex flex-col justify-center mb-24 pt-20 lg:pt-0" >

Parallel & Perpendicular Lines

Parallel and perpendicular line questions appear on almost every SAT. The core skill is: identify the slope of the given line, apply the parallel or perpendicular slope rule, then write the new equation through a given point.

The Two Slope Rules

Parallel Lines

m₁ = m₂

Same slope, different y-intercept

  • Lines never intersect — they run side by side
  • Same slope guarantees they won't cross
  • If y-intercepts also matched, the lines would be identical
y = 3x + 1 ∥ y = 3x − 7 (both slope = 3)

Perpendicular Lines

m₁ × m₂ = −1

Negative reciprocal slopes

  • Lines meet at a 90° angle
  • Rule: flip the fraction and change the sign
  • A horizontal line (slope 0) is ⊥ to a vertical line (undefined slope)
y = x + 1 ⊥ y = x + 5
{/* Negative Reciprocal Examples */}

Finding Perpendicular Slopes: Worked Examples

Original Slope Step 1: Flip the fraction Step 2: Negate the sign Perpendicular Slope
2 (= 2÷1)
−5 (flip) (negate negative)
0 (horizontal) Undefined (vertical)
{/* Full Problem Worked Examples */}

Complete Problem: Writing the Equation

Example 1: Find the line parallel to y = 4x − 3 passing through (2, 5)

Parallel → same slope: m = 4

Use point-slope: y − 5 = 4(x − 2)

y − 5 = 4x − 8

y = 4x − 3  ← Wait, same as original! Confirm: passes through (2, 5): 5 = 8 − 3 = 5 ✓

Example 2: Find the line perpendicular to 2x + 3y = 12 passing through (4, 1)

First, find slope of 2x + 3y = 12: y ={" "} x + 4, so m =

Perpendicular slope: flip and negate → m ={" "}

Point-slope: y − 1 = (x − 4)

y = x − 6 + 1

y = x − 5

SAT Trap: Parallel Lines Must Have Different Intercepts

Parallel lines need the same slope but a{" "} different y-intercept. If the intercepts also match, the lines are identical — infinitely many intersections, not parallel. The SAT sometimes includes a "same slope, same intercept" option to trap students.

{/* Section 2: Quiz */}
{ sectionsRef.current[1] = el; }} className="min-h-screen flex flex-col justify-center" >

Practice Time

{LINEAR_PARALLEL_PERP_QUIZ_DATA.map((quiz, idx) => (
))}

Topic Mastered!

); }; export default LinearParallelPerpendicularLesson;