On a blank sheet of paper, sketch △ GHI. Then sketch JK parallel to GH such that point J is on GI and point K is on HI.

Question

Explain how your proof in demonstrates that JK divides two sides of △ GHI
proportionally.

Answer 1

To sketch \( \triangle GHI \):

1. Draw an arbitrary triangle \( GHI \) where:

   • Point \( G \) is located at the top left.
   • Point \( H \) is at the top right.
   • Point \( I \) is at the bottom center.

2. Next, draw a line segment \( JK \) parallel to \( GH \):

   • Point \( J \) should be located on side \( GI \).
   • Point \( K \) should be located on side \( HI \).

Now, let’s establish how \( JK \) divides the two sides of \( \triangle GHI \) proportionally.

Proof of Proportionality:

1. Parallel Lines: By definition, a line segment that is parallel to one side of a triangle divides the other two sides proportionally. In this example, the line segment \( JK \) is parallel to side \( GH \).

2. Using the Triangle Proportionality Theorem: According to the Triangle Proportionality Theorem, if a line is drawn parallel to one side of a triangle (in this case \( GH \)), it divides the other two sides (which are \( GI \) and \( HI \)) into segments that are proportional.

   • Let’s assign the following labels:
     • Let \( GJ \) be the segment of side \( GI \).
     • Let \( JI \) be the remaining segment of side \( GI \) such that \( GI = GJ + JI \).
     • Let \( HK \) be the segment of side \( HI \).
     • Let \( KI \) be the remaining segment of side \( HI \) such that \( HI = HK + KI \).

3. Proportional Segments: According to the theorem: \[ \frac{GJ}{JI} = \frac{HK}{KI} \] This shows that the ratios of the lengths of the divided segments on sides \( GI \) and \( HI \) are equal due to the parallel line \( JK \).

4. Conclusion: Therefore, it is demonstrated that line \( JK \) divides sides \( GI \) and \( HI \) proportionally. The key to this proof lies in the fact that \( JK \) is parallel to side \( GH \), which allows the use of the Triangle Proportionality Theorem.

This proportional relationship can be further validated by measuring the lengths in a geometric setup or using similar triangles.