Triangle Decomposition of Polygons Quick Check

Question

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Question
Use the image to answer the question.

An illustration shows two pentagons. The pentagon on the left is labeled figure 1, and the pentagon on the right is labeled figure 2. Starting at the bottom left vertex of both figures, the vertices are labeled A, B, C, D, and E. In figure 1, lines connect the vertices A and C and C and E. In figure 2, lines connect the vertices A and D, B and E, and B and D.

Choose the correct decomposition of the regular polygon into n−2 triangles.

(1 point)
Responses

Figure 1 is the correct decomposition because Figure 1 is decomposed into triangles that are equal sizes.
Figure 1 is the correct decomposition because Figure 1 is decomposed into triangles that are equal sizes.

Figure 2 is the correct decomposition because Figure 2 is decomposed into 5 triangles.
Figure 2 is the correct decomposition because Figure 2 is decomposed into 5 triangles.

Figure 1 is the correct decomposition because Figure 1 is decomposed into n−2=5−2=3 triangles.
Figure 1 is the correct decomposition because Figure 1 is decomposed into n minus 2 equals 5 minus 2 equals 3 triangles.

Figure 2 is the correct decomposition because Figure 2 is decomposed into n−2 triangles.

Answer 1

To determine the correct decomposition of the polygons based on the provided options, we need to analyze the decompositions of both figures:

1. Understanding Decomposition: For a polygon with \( n \) sides, it can be decomposed into \( n-2 \) triangles. For a pentagon (5-sided polygon), this means it can be decomposed into \( 5 - 2 = 3 \) triangles.

2. Analyzing Figure 1: The description states that lines connect points A to C and C to E. If this results in 3 triangles, then it satisfies the condition of a correct decomposition for a pentagon.

3. Analyzing Figure 2: The lines connecting A to D, B to E, and B to D suggest a different arrangement. If this results in 5 triangles, then this would be more than \( n - 2 \) for a pentagon.

Now, let's evaluate the options based on this understanding:

• Figure 1 is the correct decomposition because Figure 1 is decomposed into triangles that are equal sizes.

  • This is incorrect because equal size is not a necessary condition for decomposition; it just needs to be \( n - 2 \) triangles.

• Figure 2 is the correct decomposition because Figure 2 is decomposed into 5 triangles.

  • This is incorrect; it needs to have \( n - 2 = 3 \) triangles, not 5.

• Figure 1 is the correct decomposition because Figure 1 is decomposed into \( n-2=5-2=3 \) triangles.

  • This is correct; it properly reflects the requirement.

• Figure 2 is the correct decomposition because Figure 2 is decomposed into \( n-2 \) triangles.

  • This is incorrect; as explained, it decomposed into more than \( n - 2 \) triangles.

The best response based on the analysis is:

Figure 1 is the correct decomposition because Figure 1 is decomposed into \( n-2=5-2=3 \) triangles.