Question

In the proof of the Basic Proportionality theorem for \(\triangle ABC\) with \(DE \parallel BC\) (on AB, E on AC), the construction made is to join ________.

• A. AD and AE
• B. BD and CE
• C. BE and CD
• D. AB and DE

Hint:

Visualize the triangle with the parallel line \(DE\).
The vertices to cross-connect are the lower vertices of the smaller upper triangle (\(D\) and \(E\)) with the lower vertices of the main triangle (\(C\) and \(B\)).
This forms a 'cross' shape in the lower trapezoid part: \(BE\) and \(CD\).

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
This question is from Geometry, specifically concerning the proof of the Basic Proportionality Theorem (BPT), also known as Thales's Theorem.
We need to identify the specific construction step performed in the standard geometric proof of this theorem.

Step 2: Key Formula or Approach:
Basic Proportionality Theorem states that if a line is drawn parallel to one side of a triangle intersecting the other two sides, then it divides the two sides in the same ratio.
To prove this, we construct helper triangles on the same base and between the same parallel lines to relate their areas.

Step 3: Detailed Explanation:
In a triangle \(\triangle ABC\), let a line \(DE\) be parallel to \(BC\) such that \(D\) lies on \(AB\) and \(E\) lies on \(AC\).
To prove \(\frac{AD}{DB} = \frac{AE}{EC}\), we proceed with the standard proof which relies on comparing areas of triangles.
We require triangles that share heights and have bases on the line segments \(AB\) and \(AC\).
The standard construction steps are:
1. Draw perpendiculars from \(D\) to \(AC\) (let it be \(DM \perp AC\)) and from \(E\) to \(AB\) (let it be \(EN \perp AB\)).
2. Join the vertices \(B\) to \(E\) and \(C\) to \(D\).
By joining \(B\) to \(E\), we form \(\triangle BDE\). By joining \(C\) to \(D\), we form \(\triangle CDE\).
These two triangles sit on the same base \(DE\) and between the same parallel lines \(DE\) and \(BC\), allowing us to assert that their areas are equal.
Thus, the essential segments constructed are BE and CD.

Step 4: Final Answer:
The construction made is to join BE and CD.