Question

State and prove Basic Proportionality Theorem.

Approach

The Basic Proportionality Theorem (Thales' Theorem) states: If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio. The proof involves constructing altitudes and using area ratios to establish the proportionality.

Step-by-step working

1
Statement: If DE $∥$ BC in $△ A BC$ , then $\frac{A D}{D B} = \frac{A E}{EC}$ .
1 mark
2
Given, To Prove, Construction, and Figure: Clearly state the hypothesis, conclusion, draw appropriate construction lines (e.g., join BE and CD), and provide a labeled diagram.
2 marks
3
Proof: Use area relationships. $\frac{a r ( △ A D E )}{a r ( △ B D E )} = \frac{A D}{D B}$ and $\frac{a r ( △ A D E )}{a r ( △ C D E )} = \frac{A E}{EC}$ . Since DE $∥$ BC, triangles BDE and CDE have equal areas (same base DE, equal heights). Thus the ratios are equal.
2 marks

Concepts used

basic proportionality theoremthales theoremprovesimilar trianglesmedians

Approach

Step-by-step working

Concepts used

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