Worked example 1
Question: Find the distance between A(−2,3) and B(4,−5).
Solution: AB=√[(4+2)²+(−5−3)²]=√(36+64)=√100=10 units.
What this chapter covers
Use coordinates to calculate the distance between two points and divide a line segment internally in a given ratio.
Distance formula
The distance formula follows from Pythagoras: horizontal change is x₂−x₁ and vertical change is y₂−y₁. Squaring removes sign concerns. Use it to test equal sides or classify a triangle.
Section formula
If P divides A(x₁,y₁) and B(x₂,y₂) internally in ratio m:n (AP:PB=m:n), weight A by n and B by m. The midpoint is the special case m=n=1.
Key formulae: AB=√[(x₂−x₁)²+(y₂−y₁)²]. P=( (mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n) ).
Worked examples: CBSE question patterns
These are original practice questions modeled on common board-exam patterns; they are not reproduced past-paper questions.
Worked example 2
Question: Find the point dividing A(1,2) and B(7,8) internally in ratio 2:1.
Solution: P=((2×7+1×1)/3,(2×8+1×2)/3)=(15/3,18/3)=(5,6).
Practical activity / investigation
Make a coordinate grid on squared paper. Plot a school, home and bus stop as points. Use the distance formula to compare two routes and the section formula to mark a meeting point that divides a route in a chosen ratio.
Record: Write the aim, materials, steps, observations and conclusion in your notebook. Follow teacher supervision for any classroom activity.
Quick practice with answers
Try each question before opening its solution.
Practice 1: Find the midpoint of (−4,6) and (8,−2).
Midpoint=((−4+8)/2,(6−2)/2)=(2,2).
Practice 2: Find the distance between (1,1) and (4,5).
√[(3)²+(4)²]=5 units.
Practice 3: Divide (0,0) and (12,6) in ratio 1:2 internally.
P=((1×12+2×0)/3,(1×6+2×0)/3)=(4,2).
Revision checklist
- Can I explain the key idea without memorising a worked solution?
- Have I selected the right formula and written units where needed?
- Can I show each algebraic or geometric step clearly?
- Can I check my answer by substitution, estimation or a diagram?
Aligned to the Class X Mathematics curriculum for 2026–27 and the NCERT Mathematics textbook. Always confirm any school-specific internal assessment with your teacher.