Triangle Decomposition of Polygons Quick Check

Question

2 of 52 of 5 Items

Question
Use the image to answer the question.

An illustration shows two square construction signs. Starting with the vertex on the left, the vertices are labeled A, B, C, and D. The sign on the left is labeled figure 1, and the sign on the right is labeled figure 2. In figure 1, lines connect the vertices A and C and B and D. In figure 2, a line connects the vertices B and D. An image on both signs shows a silhouette of a person holding a shovel.

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

(1 point)
Responses

Figure 1 is the correct decomposition because it has 4 triangles of the equal size.
Figure 1 is the correct decomposition because it has 4 triangles of the equal size.

Figure 2 is the correct decomposition because it has n−2=4−2=2 triangles.
Figure 2 is the correct decomposition because it has n minus 2 equals 4 minus 2 equals 2 triangles.

Figure 1 is the correct decomposition because Figure 1 is decomposed into 2 triangles that are equal sizes.
Figure 1 is the correct decomposition because Figure 1 is decomposed into 2 triangles that are equal sizes.

Figure 1 is the correct decomposition because Figure 1 is decomposed into 4 triangles that are not equal sizes.

Answer 1

To determine the correct decomposition of the polygon (which is a square in this case) into \( n-2 \) triangles, we first need to know that in a polygon with \( n \) vertices, the number of triangles obtained through triangulation is \( n-2 \).

Given that the polygon has 4 vertices (A, B, C, D), we can substitute \( n = 4 \) into \( n-2 \):

\[ n - 2 = 4 - 2 = 2 \]

This means the polygon can be decomposed into 2 triangles.

Analyzing the Figures:

• Figure 1: If Figure 1 is showing the polygon decomposed into 4 triangles, that would not satisfy the requirement of \( n-2 \) for a square. Although it could show equal-sized triangles, it is still incorrect since we only need 2 triangles.

• Figure 2: If Figure 2 connects vertices B and D (and possibly forms the other necessary line to complete the triangles), and indeed it results in 2 triangles, then it would satisfy \( n-2 \) since 4 vertices yield 2 triangles.

Conclusion:

Therefore, the correct response is:

Figure 2 is the correct decomposition because it has \( n-2 = 4-2 = 2 \) triangles.