Question

Use the image to answer the question. Choose the correct decomposition of the polygon into n−2 triangles. (1 point) Responses 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 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 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. Skip to navigation

Answer 1

To answer the question regarding the correct decomposition of a polygon into \( n - 2 \) triangles, we need to first identify the total number of sides \( n \) of the polygon. Since the specifics about the images (Figure 1 and Figure 2) are not provided, let's break down what we know based on the information given:

1. Understanding \( n - 2 \): The formula \( n - 2 \) applies to any polygon (with \( n \) sides) when decomposing it into triangles. For example, a quadrilateral (4 sides) would decompose into \( 4 - 2 = 2 \) triangles.

2. Figures: Based on the responses:

   • Figure 1 claims to have 4 triangles that may or may not be equal in size.
   • Figure 2 claims to have \( n - 2 = 2 \) triangles if \( n = 4 \).

3. Identify the correct decomposition:

   • If the polygon is a quadrilateral (which has 4 sides), then the correct decomposition into triangles would indeed result in 2 triangles.
   • If Figure 1 has 4 triangles, then it seems to incorrectly imply that 4 triangles can be a valid decomposition for a polygon with only 4 sides, which violates the \( n - 2 \) rule.

Based on this reasoning, the correct response would depend on the specific figures and should ideally reflect the triangle count as per the formula applied correctly.

If we assume the polygon mentioned has 4 sides:

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

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Answer 2

se the image to answer the question. Is this the correct decomposition of a honeycomb shaped like a hexagon into n−2 triangles? (1 point) Responses 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. 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. Yes, this is the correct decomposition because the decomposition is 6 triangles. Yes, this is the correct decomposition because the decomposition is 6 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. Skip to navigation

Answer 3

To determine the correct decomposition of a hexagonal honeycomb into \( n - 2 \) triangles, we need to follow these steps:

1. Identify the number of sides \( n \): A hexagon has 6 sides, so \( n = 6 \).

2. Calculate \( n - 2 \): \[ n - 2 = 6 - 2 = 4 \] This means that a valid decomposition of a hexagon into triangles should result in exactly 4 triangles.

3. Analyze the provided responses:

   • The response "Yes, this is the correct decomposition because the decomposition has \( n - 2 = 6 - 2 = 4 \) triangles." This response is accurate if the decomposition shown indeed results in 4 triangles.
   • The response "Yes, this is the correct decomposition because the decomposition is 6 triangles." This is incorrect, as a hexagonal decomposition yielding 6 triangles does not follow the \( n - 2 \) rule for the hexagon.
   • Responses stating "No, this is not the correct decomposition because the decomposition is 4 equal-sized triangles." are misleading. The size of the triangles matters less than the quantity. If there are 4 triangles regardless of their sizes, the decomposition is correct as per the requirement.

Conclusion:

If the decomposition clearly shows 4 triangles (whether equal or not):

• The correct choice would be:
  "Yes, this is the correct decomposition because the decomposition has \( n-2=6-2=4 \) triangles."

If the decomposition instead shows 6 triangles, then it would not be the correct answer. Ensure that the solution matches the \( n-2 \) triangle count precisely to select the correct answer.

Answer 4

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 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°. The sum of the angles of a square is 1,440 degrees . The sum of the angles of a square is 360°.

Answer 5

To derive the formula for the sum of the interior angles of a regular polygon and test it with a square, we can use the process of decomposition and the known properties of triangles.

Deriving the Formula

1. Understanding the Polygon: Let’s consider a regular polygon with \( n \) sides (where \( n \geq 3 \)).

2. Decomposing the Polygon: A polygon can be divided into triangles by drawing diagonals from one vertex to the other non-adjacent vertices. Specifically, from one vertex, we can draw \( n - 3 \) diagonals that each create a triangle—in total, we can create \( n - 2 \) triangles.

3. Calculating the Angle Sum: Each triangle has a sum of interior angles equal to \( 180^\circ \). Therefore, for \( n - 2 \) triangles: \[ \text{Total interior angle sum} = (n - 2) \times 180^\circ \]

From this, we can conclude that the sum of the interior angles of a regular polygon with \( n \) sides is given by: \[ \text{Sum of interior angles} = (n - 2) \times 180^\circ \]

Testing the Formula with a Square

Now, let’s apply this formula to find the sum of the interior angles of a square. A square is a polygon with:

• \( n = 4 \) sides.

Using the formula: \[ \text{Sum of interior angles} = (4 - 2) \times 180^\circ = 2 \times 180^\circ = 360^\circ \]

Conclusion

The sum of the angles of a square is:

• 360°.

This matches the expected interior angle sum for a square, confirming that our derived formula is correct.

Thus, among the responses given:

• The correct answer is: "The sum of the angles of a square is 360°."

Answer 6

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, 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−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, n triangles are formed, where n is the number of 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. Skip to navigation

Answer 7

When a regular polygon with \( n \) sides is decomposed into triangles using one of its vertices (i.e., by drawing diagonals from that vertex to the other non-adjacent vertices), we can determine how many triangles are formed:

1. Decomposing the Polygon:

   • Select a single vertex of the polygon.
   • From that vertex, draw diagonals to each of the other non-adjacent vertices.

2. Counting the Triangles:

   • For a polygon with \( n \) sides, you can connect the selected vertex to \( n - 3 \) other vertices (since you cannot connect it to itself or its two adjacent vertices).
   • This creates \( n - 2 \) triangles in total.

   The first triangle is formed between the selected vertex and the two adjacent vertices. Each additional diagonal you draw creates an additional triangle.

Final Conclusion

Thus, when a regular polygon with \( n \) sides is decomposed into triangles using the same vertex, the number of triangles formed is: \[ n - 2 \]

Correct Response

The appropriate choice from your responses 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."