Use the image to answer the question.

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 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 2 is the correct decomposition because Figure 2 is decomposed into n−2 triangles.
Figure 2 is the correct decomposition because Figure 2 is decomposed into n minus 2 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 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.
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Answer 1

To determine the correct decomposition of the pentagon into triangles, we first need to recognize that for a polygon with \( n \) sides (in this case, \( n = 5 \) for a pentagon), it can be divided into \( n-2 \) triangles. Therefore, for a pentagon, it can be decomposed into \( 5 - 2 = 3 \) triangles.

Based on the information provided in the question:

• Figure 1 connects vertices A and C, and C and E, which results in a decomposition into 3 triangles.
• Figure 2 connects A and D, B and E, and B and D, which results in a decomposition into 5 triangles.

Since the goal is to decompose the pentagon into \( n - 2 \) triangles, which equals 3 for this problem, Figure 1 is the correct decomposition.

Therefore, the correct response is:

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.

Answer 2

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

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 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.
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Answer 3

To determine the correct decomposition of the square (which has 4 sides, \( n = 4 \)) into triangles, we apply the formula for the number of triangles in any \( n \)-sided polygon, which is \( n - 2 \).

For a square:

• \( n - 2 = 4 - 2 = 2 \)

This means a square can be decomposed into 2 triangles.

Now, let’s evaluate the figures based on the provided details:

• Figure 1 connects vertices A and C and B and D. This implies that it uses both the diagonals of the square, creating 4 triangles from the two diagonal lines intersecting at the center.
• Figure 2 only connects B and D (the diagonal), which would logically create only 2 triangles.

Since the correct decomposition must yield \( n - 2 \) triangles for a square (which is 2), Figure 2 is the correct decomposition according to the formula and its description.

Therefore, the correct response is:

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.

Answer 4

Use the image to answer the question.

An illustration shows a bee on top of a square honeycomb. The cells inside the honeycomb are shaped like hexagons. In one of the cells, lines connect the vertex at the top left to the three non-adjacent vertices.

Is this the correct decomposition of a honeycomb shaped like a hexagon into n−2 triangles?

(1 point)
Responses

Yes, this is the correct decomposition because the decomposition is 6 triangles.
Yes, this is the correct decomposition because the decomposition is 6 triangles.

Yes, this is the correct decomposition because the decomposition has n−2=6−2=4 triangles.
Yes, this is the correct decomposition because the decomposition has n minus 2 equals 6 minus 2 equals 4 triangles.

No, this is not the correct decomposition because the decomposition is 4 equal-sized triangles.
No, this is not the correct decomposition because the decomposition is 4 equal-sized triangles.

No, this is not the correct decomposition because the decomposition is 4 triangles.
No, this is not the correct decomposition because the decomposition is 4 triangles.
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Answer 5

To determine whether the decomposition of a hexagon-shaped honeycomb into triangles is correct, we use the formula \( n - 2 \), where \( n \) is the number of sides of the polygon.

In the case of a hexagon:

• \( n = 6 \)
• Therefore, \( n - 2 = 6 - 2 = 4 \).

This means a hexagon can be decomposed into 4 triangles.

Given the description that lines connect the vertex at the top left to the three non-adjacent vertices, it indicates a possible decomposition of the hexagon into triangles. If it results in 4 triangles, that would indeed match the expected decomposition.

Now, analyzing the responses:

• One option states that the decomposition has 6 triangles, which is incorrect based on our analysis.
• Another option confirms that it has \( n - 2 = 4 \) triangles, which is correct.
• The other options discuss 4 equal-sized triangles and simply affirm that there are 4 triangles, but they do not address whether that is the expected result based on the formula.

Since the decomposition of the hexagon yields 4 triangles, which matches \( n - 2 \), the correct response is:

Yes, this is the correct decomposition because the decomposition has n−2=6−2=4 triangles. Yes, this is the correct decomposition because the decomposition has n minus 2 equals 6 minus 2 equals 4 triangles.

Answer 6

Use the angle sum of triangles to develop the interior angle sums of regular polygons formula using decomposition. Test the formula by finding the sum of the interior angles of a square.(1 point)
Responses

The sum of the angles of a square is 360°.
The sum of the angles of a square is 360 degrees .

The sum of the angles of a square is 1,080°.
The sum of the angles of a square is 1,080 degrees .

The sum of the angles of a square is 182°.
The sum of the angles of a square is 182 degrees .

The sum of the angles of a square is 1,440°.

Answer 7

To find the sum of the interior angles of a regular polygon using decomposition into triangles, we can use the formula:

Sum of interior angles = (n - 2) × 180°

where \( n \) is the number of sides of the polygon.

Now, let's apply this formula to a square:

• A square has \( n = 4 \) sides.
• Plugging this into the formula, we have:

\[ \text{Sum of interior angles} = (4 - 2) \times 180° \] \[ = 2 \times 180° \] \[ = 360° \]

Thus, the sum of the angles of a square is 360 degrees.

Therefore, the correct response is:

The sum of the angles of a square is 360°. The sum of the angles of a square is 360 degrees.

Answer 8

When a regular polygon with n sides is decomposed into triangles using the same vertex, how many triangles are formed?(1 point)
Responses

When a regular polygon is decomposed, n+2 triangles are formed, where n is the number of sides of the regular polygon. There are two more triangles than the number of the sides of the regular polygon.
When a regular polygon is decomposed, n plus 2 triangles are formed, where n is the number of sides of the regular polygon. There are two more triangles than the number of the sides of the regular polygon.

When a regular polygon is decomposed, n−2 triangles are formed, where n is the number of sides of the regular polygon. There are two fewer triangles than the number of the sides of the regular polygon.
When a regular polygon is decomposed, n minus 2 triangles are formed, where n is the number of sides of the regular polygon. There are two fewer triangles than the number of the sides of the regular polygon.

When a regular polygon is decomposed, 2⋅n triangles are formed, where n is the number of sides of the regular polygon. There are twice as many triangles formed than the number of the sides of the regular polygon.
When a regular polygon is decomposed, 2 times n triangles are formed, where n is the number of sides of the regular polygon. There are twice as many triangles formed than the number of the sides of the regular polygon.

When a regular polygon is decomposed, n triangles are formed, where n is the number of sides of the regular polygon.

Answer 9

When a regular polygon with \( n \) sides is decomposed into triangles by drawing diagonals from one vertex to the other non-adjacent vertices, the number of triangles formed is given by the formula:

\[ \text{Number of triangles} = n - 2 \]

This is because selecting one vertex and drawing lines to the other non-adjacent vertices effectively divides the polygon into \( n - 2 \) triangles.

Therefore, the correct response is:

When a regular polygon is decomposed, n−2 triangles are formed, where n is the number of sides of the regular polygon. There are two fewer triangles than the number of the sides of the regular polygon.