Distance Between Two Points

• Derive and apply the distance formula from Pythagoras' theorem.
• Find distance between A(x₁, y₁) and B(x₂, y₂): d = √[(x₂−x₁)² + (y₂−y₁)²].
• Verify types of triangles (scalene, isosceles, equilateral) using side lengths.
• Show points are collinear using distance equality or zero area method.
• Solve NCERT Ex. 3.3 applications.

1. Distance formula — d(A, B) = √[(x₂ − x₁)² + (y₂ − y₁)²].
2. Derivation — Horizontal difference and vertical difference form legs of right triangle; hypotenuse = distance.
3. Origin distance — From (x, y) to (0, 0): √(x² + y²).
4. Horizontal/vertical — If y₁ = y₂, d = |x₂ − x₁|; if x₁ = x₂, d = |y₂ − y₁|.
5. Isosceles triangle — Two equal sides among three computed distances.
6. Equilateral — All three distances equal.
7. Collinearity test — Largest distance = sum of other two (for points on line segment).
8. NCERT Example 3.4 — Distance between (2, 3) and (4, 1).
9. Units — Distance has same unit as coordinates (cm, km, etc.).
10. Non-negativity — Distance is always ≥ 0; equals 0 only when points coincide.

NCERT: Find distance between P(2, −3) and Q(10, y) if PQ = 10

Step 1 — PQ² = (10 − 2)² + (y − (−3))² = 10²
Step 2 — 64 + (y + 3)² = 100
Step 3 — (y + 3)² = 36 → y + 3 = ±6
Step 4 — y = 3 or y = −9
Both values satisfy the given distance

• Forgetting square root at end (leaving d²).
• Sign error: (y₂ − y₁) not (y₁ − y₂) squared — same result, but mixed subtraction causes arithmetic slips.
• Using d = (x₂−x₁) + (y₂−y₁) ( Manhattan distance — wrong for Euclidean).
• Not checking which y value when quadratic gives two solutions.
• Comparing squared distances for equality tests (valid shortcut) but forgetting to sqrt for final answer.

• Distance between (1, 2) and (4, 6)?
• Is triangle with vertices (0,0), (3,0), (0,4) right-angled?
• Distance of (−3, 4) from origin?
• If A(1,1) and B(4,5), find AB.
• State the distance formula.

Open the Practice tab for graded questions on Distance Between Two Points.