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Difficulty: MediumArea of Two-Dimensional Shapes

A digital designer is scaling two similar triangular logos. The smaller logo has a perimeter of 1818 centimeters and an area of 1212 square centimeters. If the larger logo has a perimeter of 5454 centimeters, what is the area, in square centimeters, of the larger logo?

  1. A
    3636
  2. B
    7272
  3. 108108Answer
  4. D
    324324

Answer

108 square centimeters
The correct answer is 108108. Since the two triangular logos are similar, the ratio of their areas is the square of the ratio of their perimeters. The ratio of their perimeters is 5418=3\frac{54}{18} = 3, meaning the larger logo has linear dimensions that are 33 times those of the smaller logo. Squaring this scale factor gives the area scaling factor, which is 32=93^2 = 9. Therefore, the area of the larger logo is 12×9=10812 \times 9 = 108 square centimeters.

Step-by-Step Solution

1
Determine the linear scale factor between the two similar logos.
The linear scale factor kk is 5418=3\frac{54}{18} = 3.
For similar figures, the ratio of any corresponding linear dimensions (such as perimeters) is equal to the linear scale factor.
2
Find the area scaling factor.
The area scaling factor is k2=32=9k^2 = 3^2 = 9.
The ratio of the areas of two similar figures is equal to the square of their linear scale factor.
3
Calculate the area of the larger logo.
The area of the larger logo is 12×9=10812 \times 9 = 108 square centimeters.
Multiplying the area of the smaller logo by the area scaling factor yields the area of the larger logo.

Key Concept

The ratio of the areas of two similar two-dimensional shapes is equal to the square of their linear scale factor.
Estimated Time:1m 30s
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Topics

Area of Two-Dimensional Shapes

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