Distance Formulas and Section Formula

• Distance between two points and its applications.
• Section formula — internal and external division.
• Point-to-line distance: |Ax₀ + By₀ + C|/√(A² + B²).
• Distance between parallel lines |C₁ − C₂|/√(A² + B²).
• Foot of perpendicular from a point to a line.

Level 1 — Distance and section formula

Distance formula: d(P₁, P₂) = √((x₂−x₁)² + (y₂−y₁)²) (Pythagorean theorem on coordinate plane).

Section formula — internal division: Point dividing P₁(x₁,y₁) and P₂(x₂,y₂) in ratio m:n internally: P = ((mx₂ + nx₁)/(m+n), (my₂ + ny₁)/(m+n)).

Section formula — external division: Dividing in ratio m:n externally (beyond P₂): P = ((mx₂ − nx₁)/(m−n), (my₂ − ny₁)/(m−n)), m ≠ n.

Midpoint (m = n = 1): M = ((x₁+x₂)/2, (y₁+y₂)/2).

Centroid of triangle (x₁,y₁), (x₂,y₂), (x₃,y₃): G = ((x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3).

Level 2 — Point-to-line and parallel line distances, foot of perpendicular

Point-to-line distance: Distance from P(x₀, y₀) to line Ax + By + C = 0: d = |Ax₀ + By₀ + C| / √(A² + B²).

Derivation idea: Write the normal from P to the line. Foot of perpendicular H lies on the line. Use vector projection: PH = (A·x₀ + B·y₀ + C)/√(A²+B²) (signed), distance = |PH|.

Distance between parallel lines Ax + By + C₁ = 0 and Ax + By + C₂ = 0: d = |C₁ − C₂| / √(A² + B²). (Derived by taking any point on one line and computing its distance to the other.)

Foot of perpendicular from P(x₀, y₀) to line Ax + By + C = 0: Let foot be H(h, k). Conditions: H lies on the line (Ah + Bk + C = 0) and PH ⊥ line. Solving: (h − x₀)/A = (k − y₀)/B = −(Ax₀ + By₀ + C)/(A² + B²). So h = x₀ − A(Ax₀ + By₀ + C)/(A² + B²), k = y₀ − B(Ax₀ + By₀ + C)/(A² + B²).

Reflection of P across line: Image P' satisfies H = midpoint of PP'. So P' = (2h − x₀, 2k − y₀) where H is foot.

JEE pattern: Locus of point equidistant from two intersecting lines is the angle bisectors. Angle bisectors of Ax+By+C₁=0 and Ax+By+C₂=0: (Ax+By+C₁)/√(A²+B²) = ±(Ax+By+C₂)/√(A²+B²).

NCERT spotlight

Area of triangle with vertices (x₁,y₁), (x₂,y₂), (x₃,y₃): Area = (1/2)|x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂)|. This is zero iff points are collinear (connects to previous note). For the perpendicular from a point to a line, the parameter t = −(Ax₀+By₀+C)/(A²+B²) gives the signed distance — key for reflection and foot computations.

Find the distance from point P(3, −4) to the line 4x − 3y + 5 = 0. Also find the foot of perpendicular.

Step 1 — Distance: d = |4(3) − 3(−4) + 5| / √(4² + 3²) = |12 + 12 + 5| / √25 = 29/5 = 5.8.

Step 2 — Foot of perpendicular H(h, k):
         t = −(Ax₀ + By₀ + C)/(A² + B²) = −(12 + 12 + 5)/25 = −29/25.
Step 3 — h = x₀ + A·t = 3 + 4·(−29/25) = 3 − 116/25 = 75/25 − 116/25 = −41/25.
         k = y₀ + B·t = −4 + (−3)·(−29/25) = −4 + 87/25 = −100/25 + 87/25 = −13/25.
Step 4 — Foot H = (−41/25, −13/25).
Step 5 — Verify H on line: 4(−41/25) − 3(−13/25) + 5 = −164/25 + 39/25 + 125/25 = 0. ✓

Find the distance between the parallel lines 3x − 4y + 7 = 0 and 3x − 4y − 8 = 0.

Step 1 — Both lines: A = 3, B = −4. C₁ = 7, C₂ = −8.
Step 2 — Distance = |C₁ − C₂| / √(A² + B²) = |7 − (−8)| / √(9 + 16) = 15/5 = 3.
Step 3 — Alternatively: find any point on first line, e.g., (−1, 1) [check: 3(−1)−4(1)+7=0 ✓].
         Distance from (−1,1) to 3x−4y−8=0: |3(−1)−4(1)−8|/5 = |−3−4−8|/5 = 15/5 = 3. ✓

1. Find the distance between points (−3, 4) and (5, −2).
2. Find the coordinates of the point dividing (1, 3) and (5, 11) in ratio 3:1 internally.
3. Find the distance from (0, 0) to the line 5x + 12y − 65 = 0.
4. Find the distance between parallel lines 2x + 3y − 4 = 0 and 2x + 3y + 5 = 0.
5. Stretch: Find the reflection of point (1, 2) across the line x + y − 1 = 0.

Open the Practice tab for graded questions on Straight Lines — Distance.

• Use the platform-native live simulation or PhET-style tool for this topic (number line, Venn, physics playground, molecule builder, sensor dashboard, etc.).
• Mirror / body / home activity: physically do the concept (count objects, measure, role-play) and photograph or describe for portfolio.
• Voice or text reflection with AI Mentor: explain the concept to a younger student or family member.

• "Explain this concept to a Class 6 student using one real example from an Indian home, school, market, or festival."
• "What is one common mistake students make here, and how would you catch yourself making it?"
• Stretch: "How does this connect to coding, robotics, money, health, environment, or a future career?"

• Complete the core practice + one extension activity (photo, table, short reflection, or mini-project) for base XP + topic badge.
• 5-7 day streak or family discussion note = multiplier + visible artifact in parent/principal dashboard.
• Best real-world application stories (anonymised) featured on class or national leaderboard.

• One hands-on project or measurement using the Drishti kit or household items that makes the concept physical.
• Direct link to at least one Future Skill track (Money Management, Green Tech, Cyber Defenders, Micro-Entrepreneurship, AI Mastery, Sustainable Living, Personality Development).
• Coding extension where relevant (simple script, simulation, or data logging).

This material emphasises experiential "learning by doing", competency (apply/create/analyse), vocational exposure, critical thinking, and multidisciplinary connections. Designed to feed live worlds, AI Mentor (with memory), gamification, robotics, parent analytics, and future skills — not just exam prep.

Portfolio Evidence Idea: Your photo/table/reflection/project + one sentence on "How this helps me in real life or a possible future path."

Open the Practice tab for aligned questions (easy/medium/hard + case-based) with full AI scaffolding.

See curriculum for cross-links and the full future-skills/robotics chapters.