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Answer :
To find the area of the larger pentagon, we need to use the scale factor between the two similar pentagons and the area of the smaller pentagon.
1. Understand the Scale Factor:
- The scale factor between the two pentagons is given as 4:5. This means for every 4 units of length in the smaller pentagon, the corresponding length in the larger pentagon is 5 units.
2. Area Relationship in Similar Figures:
- For similar figures, the ratio of their areas is the square of the ratio of their corresponding side lengths. Therefore, the ratio of the areas of the smaller to the larger pentagon is [tex]\((4/5)^2\)[/tex].
3. Calculate the Area of the Larger Pentagon:
- We know the area of the smaller pentagon is 120 cm².
- The ratio of the areas using the scale factor is [tex]\((4/5)^2 = 16/25\)[/tex].
- If the area of the larger pentagon is [tex]\(A\)[/tex], then the equation to find [tex]\(A\)[/tex] is:
[tex]\[
\frac{120}{A} = \frac{16}{25}
\][/tex]
- Cross-multiply to solve for [tex]\(A\)[/tex]:
[tex]\[
120 \times 25 = 16 \times A
\][/tex]
[tex]\[
3000 = 16A
\][/tex]
[tex]\[
A = \frac{3000}{16} = 187.5
\][/tex]
The area of the larger pentagon is 187.5 cm².
Since we only referred to pentagons and not octagons in this problem, it's clear that we were working with pentagons throughout. There is no octagon mentioned elsewhere in the problem, so it's possible the question had an error regarding the shape mentioned.
1. Understand the Scale Factor:
- The scale factor between the two pentagons is given as 4:5. This means for every 4 units of length in the smaller pentagon, the corresponding length in the larger pentagon is 5 units.
2. Area Relationship in Similar Figures:
- For similar figures, the ratio of their areas is the square of the ratio of their corresponding side lengths. Therefore, the ratio of the areas of the smaller to the larger pentagon is [tex]\((4/5)^2\)[/tex].
3. Calculate the Area of the Larger Pentagon:
- We know the area of the smaller pentagon is 120 cm².
- The ratio of the areas using the scale factor is [tex]\((4/5)^2 = 16/25\)[/tex].
- If the area of the larger pentagon is [tex]\(A\)[/tex], then the equation to find [tex]\(A\)[/tex] is:
[tex]\[
\frac{120}{A} = \frac{16}{25}
\][/tex]
- Cross-multiply to solve for [tex]\(A\)[/tex]:
[tex]\[
120 \times 25 = 16 \times A
\][/tex]
[tex]\[
3000 = 16A
\][/tex]
[tex]\[
A = \frac{3000}{16} = 187.5
\][/tex]
The area of the larger pentagon is 187.5 cm².
Since we only referred to pentagons and not octagons in this problem, it's clear that we were working with pentagons throughout. There is no octagon mentioned elsewhere in the problem, so it's possible the question had an error regarding the shape mentioned.
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