Mathematics · Triangles · NCERT Examples

Example 1

A complete, independently verified solution for NCERT Example 1.

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Mathematics · Chapter 6 · NCERT Examples

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NCERT Example

Example 1

basic proportionality theorem · ratios · parallel lines

A line intersects sides AB and AC of △ABC at D and E respectively and is parallel to BC. Prove that AD/AB = AE/AC.

Triangle ABC with D on AB, E on AC, and line DE parallel to BC.
Figure 6-13 from the supplied NCERT chapter.

Solution

Given:

In triangle ABC, D lies on AB, E lies on AC and DE is parallel to BC.

To prove:

AD/AB = AE/AC.

Triangle ABC with D on AB, E on AC, and line DE parallel to BC.
Figure 6-13 from the supplied NCERT chapter.

Proof:

By the Basic Proportionality Theorem:

ADDB=AEEC\frac{AD}{DB}=\frac{AE}{EC}

Taking reciprocals:

DBAD=ECAE\frac{DB}{AD}=\frac{EC}{AE}

Adding 1 to both sides:

DBAD+1=ECAE+1\frac{DB}{AD}+1=\frac{EC}{AE}+1
DB+ADAD=EC+AEAE\frac{DB+AD}{AD}=\frac{EC+AE}{AE}
ABAD=ACAE\frac{AB}{AD}=\frac{AC}{AE}

Taking reciprocals again:

ADAB=AEAC\frac{AD}{AB}=\frac{AE}{AC}

Final answer

Therefore, AD/AB = AE/AC.

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