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Compare the perimeter and area of the original figure to the enlarged figure using proportions.

A small rectangle has a length of 2 and width of 1. A large rectangle has a length of b and width of 2. StartFraction enlarged side (2) over Original side (1) = StartFraction Enlarged perimeter (4 + 2 b) over Original perimeter (6) EndFraction. (StartFraction enlarged side (2) over Original side (1) EndFraction) squared = StartFraction Enlarged area (2 b) over Original area (2) EndFraction
Which statements describe true relationships? Check all that apply.
The enlarged area is 4 times the original area, which is the square of the scale factor.
The scale factor is One-half.
The enlarged perimeter is 2 times the original perimeter, which is the scale factor.
The perimeter is proportional to the scale factor and the area is proportional to the square of the scale factor.
The enlarged area and perimeter are equal.

1 answer

Let's analyze the problem step by step.

  1. Original Rectangle Dimensions:

    • Length = 2
    • Width = 1
    • Perimeter = 2 * (Length + Width) = 2 * (2 + 1) = 6
    • Area = Length * Width = 2 * 1 = 2

  2. Enlarged Rectangle Dimensions:

    • Length = b
    • Width = 2
    • Perimeter = 2 * (b + 2) = 2b + 4
    • Area = b * 2 = 2b

  3. Scale Factor:

    • We have that for an enlarged side, the relationship is given by: \[ \frac{\text{enlarged side (2)}}{\text{original side (1)}} = 2 \]
    • This implies that the scale factor (k) is 2 for the specific side being compared (from 1 to 2).

  4. Comparing Perimeter:

    • The statement provided is: \[ \frac{\text{Enlarged perimeter (4 + 2b)}}{\text{Original perimeter (6)}} \] Simplifying gives: \[ \frac{2b + 4}{6} \]
    • To check proportionality:
      • With a scale factor of \(k = 2\), we expect the enlarged perimeter to equal \(2\) times the original perimeter (\(2 \times 6 = 12\)).
      • Setting \((2b + 4) = 12\) leads to \(2b = 8 \Rightarrow b = 4\).
      • Hence, the statements about the perimeter (the scale factor applied) will hold true.

  5. Comparing Area:

    • The area relationship is given as: \[ \frac{\text{enlarged area (2b)}}{\text{original area (2)}} \]
    • We have: \[ \frac{2b}{2} = b \]
    • Since the scale factor is \(k = 2\), we square the scale factor for area: \[ k^2 = 2^2 = 4 \]
    • Thus, if \(b\) is indeed 4 (from the perimeter analysis), the enlarged area will be \(2 \times 4 = 8\), which indeed is 4 times the original area.

Now, let’s evaluate the statements based on our findings:

  • The enlarged area is 4 times the original area, which is the square of the scale factor.

    • True (as \(4 = 2^2\)).

  • The scale factor is One-half.

    • False (the scale factor is 2, not 0.5).

  • The enlarged perimeter is 2 times the original perimeter, which is the scale factor.

    • True (the enlarged perimeter is indeed twice the original perimeter, holds true when \(b = 4\)).

  • The perimeter is proportional to the scale factor and the area is proportional to the square of the scale factor.

    • True (perimeters follow the scale factor directly, areas follow the square of the scale factor).

  • The enlarged area and perimeter are equal.

    • False (the enlarged area is 8 and the perimeter is 12 when \(b = 4\)).

Based on the analysis, the following statements are true:

  1. The enlarged area is 4 times the original area, which is the square of the scale factor.
  2. The enlarged perimeter is 2 times the original perimeter, which is the scale factor.
  3. The perimeter is proportional to the scale factor and the area is proportional to the square of the scale factor.