To demonstrate that line segment JK divides the sides of triangle \( \triangle GHI \) proportionally, we can use the Basic Proportionality Theorem, also known as the Thales theorem. Here is how you can construct the sketch and provide an explanation:
Step 1: Sketch Triangle \( \triangle GHI \)
- Draw triangle \( GHI \):
- Plot points \( G \), \( H \), and \( I \) to form the vertices of the triangle.
- Connect points \( G \) to \( H \), \( H \) to \( I \), and \( I \) to \( G \).
Step 2: Draw Line Segment \( JK \)
- Draw a line segment \( JK \) such that it is parallel to side \( GH \).
- Place point \( J \) on side \( GI \).
- Place point \( K \) on side \( HI \).
Step 3: Explanation of Proportionality
According to the Basic Proportionality Theorem, if a line is drawn parallel to one side of a triangle and intersects the other two sides, then it divides those two sides proportionally.
In the context of \( \triangle GHI \):
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Identify Proportions:
- Let \( GJ \) be a segment on side \( GI \).
- Let \( HK \) be a segment on side \( HI \).
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Setting Up the Proportions:
- Since \( JK \) is parallel to \( GH \), by the theorem, we have: \[ \frac{GJ}{JI} = \frac{HK}{KH} \]
- This means that the ratio of the lengths of segment \( GJ \) to \( JI \) is equal to the ratio of the lengths of segment \( HK \) to \( KH \).
Conclusion
The placement of \( JK \) parallel to \( GH \) creates a scenario where both sides \( GI \) and \( HI \) are split into segments that maintain proportional relationships. This is a direct illustration of how corresponding segments of a triangle are divided proportionally by a line parallel to one of its sides. This proof could be further validated using specific measurements or algebraic expressions, but fundamentally relies on the geometric properties outlined in the Basic Proportionality Theorem.
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