https://file.aiquickdraw.com/mathbot/file/1726701609442mqsw447u.jpg

Let's analyze the questions on the image and work through them step by step:

1. **Determine the scale factor to confirm that \( \triangle DEF \) is a scale drawing of \( \triangle ABC \):**

   To confirm that \( \triangle DEF \) is a scale drawing, we need to check if the corresponding side lengths have a constant ratio (i.e., if the scale factor is consistent).

   - Corresponding sides are:
     - \( AB \) and \( DE \)
     - \( BC \) and \( EF \)
     - \( AC \) and \( DF \)

   From the image, the lengths are as follows:
   - \( AB = 8 \) cm, \( DE = 4 \) cm
   - \( BC = 17 \) cm, \( EF = 8.5 \) cm
   - \( AC = 21 \) cm, \( DF = 10.5 \) cm

   The scale factor is calculated by dividing the side lengths of \( \triangle DEF \) by the corresponding sides of \( \triangle ABC \):

   \[
   \text{Scale factor} = \frac{DE}{AB} = \frac{4}{8} = 0.5
   \]

   Checking the other sides:
   \[
   \frac{EF}{BC} = \frac{8.5}{17} = 0.5
   \]
   \[
   \frac{DF}{AC} = \frac{10.5}{21} = 0.5
   \]

   Since all ratios are consistent, the scale factor is \( 0.5 \), confirming that \( \triangle DEF \) is a scale drawing of \( \triangle ABC \).

2. **Calculate the areas of \( \triangle ABC \) and \( \triangle DEF \):**

   The area of a right triangle can be calculated using the formula:
   \[
   \text{Area} = \frac{1}{2} \times \text{base} \times \text{height}
   \]

   - For \( \triangle ABC \), the base is \( AC = 21 \) cm and the height is \( AB = 8 \) cm:
     \[
     \text{Area of } \triangle ABC = \frac{1}{2} \times 21 \times 8 = 84 \text{ square cm}
     \]

   - For \( \triangle DEF \), the base is \( DF = 10.5 \) cm and the height is \( DE = 4 \) cm:
     \[
     \text{Area of } \triangle DEF = \frac{1}{2} \times 10.5 \times 4 = 21 \text{ square cm}
     \]

3. **What factor should you apply to the area of \( \triangle ABC \) to find the area of \( \triangle DEF \)? How does the area of \( \triangle DEF \) compare to the area of \( \triangle ABC \)? How does this relationship compare to the scale factor?**

   Since the scale factor for the side lengths is \( 0.5 \), the area scale factor will be the square of the side length scale factor:
   \[
   \text{Area scale factor} = (0.5)^2 = 0.25
   \]

   The area of \( \triangle DEF \) should be \( 0.25 \) times the area of \( \triangle ABC \). Let's confirm:
   \[
   \text{Area of } \triangle DEF = 0.25 \times \text{Area of } \triangle ABC = 0.25 \times 84 = 21 \text{ square cm}
   \]

   This matches the area calculated earlier. Thus, the area of \( \triangle DEF \) is \( 0.25 \) times the area of \( \triangle ABC \), consistent with the square of the side length scale factor.

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### Related Questions:
1. What is the significance of the scale factor in geometric transformations?
2. How does the scale factor affect the perimeter of similar triangles?
3. Can the scale factor ever be greater than 1, and if so, what does it represent?
4. How do you verify if two triangles are similar based on angle measures?
5. How does the scale factor influence the ratio of volumes in 3D shapes?

### Tip:
Remember, when comparing areas of similar figures, the area ratio is always the square of the side length ratio!

In the figures shown, ΔDEF is a scale drawing of ΔABC. All lengths shown are in centimeters. Determine the scale factor to confirm that ΔDEF is a scale drawing of ΔABC, calculate the areas of ΔABC and ΔDEF, and what factor should be applied to the area of ΔABC to find the area of ΔDEF.

Learn how to determine the scale factor between two similar triangles, ΔABC and ΔDEF, and calculate their areas. This geometry problem covers key concepts like triangle similarity, area ratios, and scale factors, suitable for Grade 7 students.

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