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

Q: In $\Delta ABC$, $DE \parallel BC$. If $AD = x$, $DB = x - 2$, $AE = x + 2$ and $EC = x - 1$, then find the value of x.

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

Triangles

Basic Proportionality Theorem (Thales' Theorem)

Official Answer

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

Basic Proportionality TheoremThales theoremDE parallel BCsimilar trianglesratio of sidescross multiplication

Marking Scheme

11 mark: correctly stating and applying $\frac{AD}{DB} = \frac{AE}{EC}$ using the Basic Proportionality Theorem.
21 mark: correct algebraic simplification and final answer $x = 4$ .

Hint

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

Quick Oral Answer

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

Analysis & Explanation

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

Concept

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

Key points

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

Common mistakes

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

Common Mistakes

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

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

Spotted a mistake or something unclear?

Tell us — we fix reported answers fast.

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 $\frac{AD}{DB} = \frac{AE}{EC}$ and not $\frac{AD}{AB} = \frac{AE}{AC}$ used here?

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