Worked example 1
Question: In △ABC, DE∥BC. If AD=3, DB=2 and AE=4.5, find EC.
Solution: By BPT, AD/DB=AE/EC. Thus 3/2=4.5/EC, so 3EC=9 and EC=3.
What this chapter covers
Use similarity criteria, the Basic Proportionality Theorem and its converse to reason about proportional sides and unknown lengths.
Similar triangles
Similar triangles have equal corresponding angles and proportional corresponding sides. AA similarity is especially useful; SSS and SAS criteria also establish similarity. Keep vertex order consistent when writing a correspondence.
Basic Proportionality Theorem
If a line parallel to one side of a triangle intersects the other two sides, it divides those sides in the same ratio. The converse says equal division ratios imply the line is parallel to the third side. Similarity can then establish further side ratios.
Pythagoras connection
For a right triangle, the square of the hypotenuse equals the sum of the squares of the legs. Similarity provides a proof and many length problems.
Key formulae: If DE ∥ BC in △ABC, then AD/DB = AE/EC. For similar triangles, corresponding side ratios are equal.
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: Two similar triangles have corresponding sides 6 and 9. If another side of the smaller triangle is 8, find its corresponding side.
Solution: Scale factor larger/smaller=9/6=3/2. Corresponding larger side=8×3/2=12.
Practical activity / investigation
Draw a large triangle and mark points D and E on two sides. Draw DE parallel to the third side using a set square. Measure AD, DB, AE and EC; compare AD/DB with AE/EC and discuss measurement error.
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: If △ABC∼△PQR and AB/PQ=2/3 while BC=10, find QR.
BC/QR=2/3, so 10/QR=2/3 and QR=15.
Practice 2: A line divides two sides of a triangle in ratios 4:5 and 8:10. Is it parallel to the third side?
Yes: 4/5=8/10, so the converse of BPT applies.
Practice 3: A right triangle has legs 9 cm and 12 cm. Find the hypotenuse.
c²=9²+12²=225, so c=15 cm.
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.