Question 222 marksVery Short AnswerSection B

In ΔABC\Delta ABC, DE∥BCDE \parallel BC. If AD=xAD = x, DB=x−2DB = x - 2, AE=x+2AE = x + 2 and EC=x−1EC = x - 1, then find the value of x.

Triangles
Basic Proportionality Theorem (Thales' Theorem)

Official answer

x=4x = 4, found using ADDB=AEEC\frac{AD}{DB} = \frac{AE}{EC} (Basic Proportionality Theorem), verified since ADDB=42=2=63=AEEC\frac{AD}{DB} = \frac{4}{2} = 2 = \frac{6}{3} = \frac{AE}{EC}.

Answer keywords:Basic Proportionality TheoremThales theoremDE parallel BCsimilar trianglesratio of sidescross multiplication

Marking scheme

  1. 1.1 mark: correctly stating and applying ADDB=AEEC\frac{AD}{DB} = \frac{AE}{EC} using the Basic Proportionality Theorem.
  2. 2.1 mark: correct algebraic simplification and final answer x=4x = 4.

Hint

Use the Basic Proportionality Theorem: since DE∥BCDE \parallel BC, ADDB=AEEC\frac{AD}{DB} = \frac{AE}{EC}.

Quick oral answer

Since DE is parallel to BC, by the Basic Proportionality Theorem, ADDB=AEEC\frac{AD}{DB} = \frac{AE}{EC}, giving xx−2=x+2x−1\frac{x}{x-2} = \frac{x+2}{x-1}; solving this equation gives x=4x = 4.

Common mistakes

  1. 1.Setting up an incorrect ratio such as ADAB=AEAC\frac{AD}{AB} = \frac{AE}{AC} instead of the correct ADDB=AEEC\frac{AD}{DB} = \frac{AE}{EC} for the two divided segments.
  2. 2.Errors while expanding (x+2)(x−2)(x+2)(x-2) as x² - 4, sometimes mistakenly writing x2+4x^2 + 4 or x2−4xx^2 - 4x.
  3. 3.Not verifying that all computed lengths (DB=x−2,EC=x−1DB = x-2, EC = x-1) are positive for x=4x = 4, which is essential to confirm the answer is geometrically valid.

Analysis and explanation

Applies the Basic Proportionality Theorem (Thales' Theorem) since DE∥BCDE \parallel BC.


Concept

  • A line parallel to one side of a triangle divides the other two sides in the same ratio: ADDB=AEEC\frac{AD}{DB} = \frac{AE}{EC}.

Key points

  • Substituting given values: xx−2=x+2x−1\frac{x}{x-2} = \frac{x+2}{x-1}.
  • Cross-multiplying and simplifying (difference of squares on the RHS) cancels the x² terms, giving x=4x = 4.
  • Check: AD=4,DB=2,AE=6,EC=3AD=4, DB=2, AE=6, EC=3 — both ratios equal 2, confirming consistency.

Common mistakes

  • Setting up the wrong ratio (e.g., ADAB=AEEC\frac{AD}{AB} = \frac{AE}{EC} instead of ADDB=AEEC\frac{AD}{DB} = \frac{AE}{EC}).
  • Sign errors while expanding (x+2)(x−2)(x+2)(x-2).

Interesting facts

The Basic Proportionality Theorem is also known as Thales' Theorem, named after the ancient Greek mathematician Thales of Miletus (circa 600 BCE), one of the earliest recorded users of geometric reasoning.

This theorem is the converse-and-direct foundation for proving similarity of triangles by the AA (or SAS) criterion when a line is drawn parallel to one side.

The same ratio ADDB=AEEC\frac{AD}{DB} = \frac{AE}{EC} also implies ADAB=AEAC\frac{AD}{AB} = \frac{AE}{AC}, a frequently used alternate form of the same theorem in coordinate and mensuration problems.

Question details

Q22 of 56 questions

Board
CBSE
Class
Class 10
Subject
Mathematics (Standard)
Year
2026
Section
Section B
Marks
2
Type
Very Short Answer
Chapter
Triangles
Topic
Basic Proportionality Theorem (Thales' Theorem)
Bloom's level
apply
Syllabus topic
Unit 4: Geometry — Triangles: Basic Proportionality Theorem and its converse
Author
CBSE
Reviewed by
Padhantu Academic Team

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Frequently asked questions

What is the Basic Proportionality Theorem?

If a line is drawn parallel to one side of a triangle intersecting the other two sides, it divides those two sides in the same ratio.

Why is ADDB=AEEC\frac{AD}{DB} = \frac{AE}{EC} and not ADAB=AEAC\frac{AD}{AB} = \frac{AE}{AC} used here?

Both forms are valid consequences of the theorem, but since the problem gives the two separate segment lengths DB and EC directly, ADDB=AEEC\frac{AD}{DB} = \frac{AE}{EC} is the most direct ratio to use.