Question
6. A figure is dilated by a factor of 0.5 . By what factor will the area of the figure change? 0.25 0.5 1.5
Ask by Black Watkins. in the United States
Feb 03,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The area of the figure will change by a factor of 0.25.
Solution
When a figure undergoes a dilation (scaling), both its dimensions and area change based on the scale factor.
**Key Concept:**
- **Linear Scale Factor (k):** Determines how much each dimension (length, width) of the figure is multiplied.
- **Area Scale Factor:** The area changes by the square of the linear scale factor, i.e., \( \text{Area Factor} = k^2 \).
**Given:**
- **Scale Factor (k) = 0.5**
**Calculation:**
\[
\text{Area Factor} = (0.5)^2 = 0.25
\]
**Conclusion:**
The area of the figure will decrease by a factor of **0.25** when dilated by a factor of 0.5.
**Answer:**
0.25
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Simplify this solution Bonus Knowledge
When a figure is dilated by a factor, the area of the figure changes by the square of that factor. In this case, dilating by a factor of 0.5 means the area will change by a factor of \( (0.5)^2 \), which equals 0.25. So, the area of the figure will decrease to 25% of its original size! Moreover, this concept is not just a mathematical curiosity; it has real-world applications in fields like architecture and art. For instance, when scaling down models for visual presentations or blueprints, understanding how area changes helps in maintaining proportions and ensuring the designs are feasible and visually appealing.
