Question

State and prove Basic Proportionality Theorem.

Approach

State and prove the Basic Proportionality Theorem: If a line is drawn parallel to one side of a triangle, it divides the other two sides in the same ratio. Given triangle ABC with DE parallel to BC, prove $\frac{A D}{D B} = \frac{A E}{EC}$ using area method or similar triangles.

Step-by-step working

1
Statement: 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
1 mark
2
Given: Triangle ABC with DE parallel to BC. To prove: $\frac{A D}{D B} = \frac{A E}{EC}$ . Construction and figure drawn correctly
2 marks
3
Proof using area method or similar triangles: triangles ADE and BDE have same height from D, so $\frac{ar ( A D E )}{ar ( B D E )} = \frac{A D}{D B}$ . Similarly for triangles on the other side. Since DE parallel BC, corresponding triangles are equal in area ratio, leading to $\frac{A D}{D B} = \frac{A E}{EC}$
2 marks

Concepts used

basic proportionality theoremprovesimilar trianglesmedians

Approach

Step-by-step working

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