Math Problem Statement

Determine the scale factor of the following pairs of similar figures. In part (a), rectangle EFGC is a larger version of rectangle ABCD. In part (b), triangle ABC is an enlargement of triangle ADE.

Solution

To determine the scale factor for each part of this problem, let's analyze each figure separately:

Part (a)

In part (a), we have two rectangles, ABCD and EFGC. Rectangle ABCD is a smaller rectangle within EFGC.

  1. The dimensions of rectangle ABCD are given as 2cm2 \, \text{cm} by 4cm4 \, \text{cm}.
  2. The dimensions of rectangle EFGC are not directly provided, but since it appears that the larger rectangle doubles the dimensions of ABCD in each direction, we assume:
    • Height: 2×2cm=4cm2 \times 2 \, \text{cm} = 4 \, \text{cm}
    • Width: 2×4cm=8cm2 \times 4 \, \text{cm} = 8 \, \text{cm}

Thus, the scale factor from ABCD to EFGC is: Scale factor=Length of EFGCLength of ABCD=8cm4cm=2\text{Scale factor} = \frac{\text{Length of EFGC}}{\text{Length of ABCD}} = \frac{8 \, \text{cm}}{4 \, \text{cm}} = 2 So, the scale factor is 22.

Part (b)

In part (b), we have two triangles, ABC\triangle ABC and ADE\triangle ADE, with ABC\triangle ABC being an enlargement of ADE\triangle ADE.

  1. In ABC\triangle ABC, side AC=5cmAC = 5 \, \text{cm} and BC=4cmBC = 4 \, \text{cm}.
  2. In ADE\triangle ADE, which is the smaller triangle, side DEDE is assumed to be the corresponding side to BCBC in ABC\triangle ABC.

Since BC=4cmBC = 4 \, \text{cm} and DE=2cmDE = 2 \, \text{cm}: Scale factor=Length of BCLength of DE=4cm2cm=2\text{Scale factor} = \frac{\text{Length of BC}}{\text{Length of DE}} = \frac{4 \, \text{cm}}{2 \, \text{cm}} = 2 Thus, the scale factor from ADE\triangle ADE to ABC\triangle ABC is also 22.

Summary

  • Scale factor for part (a): 22
  • Scale factor for part (b): 22

Would you like further explanations on how scale factors work or have any additional questions?

Related Questions

  1. How would the scale factor change if only one dimension of the figures doubled?
  2. What happens to the area of a shape if the scale factor is doubled?
  3. Can you determine the scale factor if only the areas of two similar figures are given?
  4. How would you calculate the scale factor if given the perimeters of similar figures?
  5. What is the impact of the scale factor on the angles of similar figures?

Tip

When working with similar figures, remember that corresponding sides are proportional, but angles remain unchanged.

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Math Problem Analysis

Mathematical Concepts

Scale Factor
Similar Figures
Proportions

Formulas

Scale factor = Length of larger figure / Length of smaller figure

Theorems

Properties of Similar Figures

Suitable Grade Level

Grades 7-9