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Answer :
The ratio of the lengths of the corresponding sides is [tex]$\sqrt[2]{9}$[/tex]or [tex]$3$[/tex], and the ratio of the perimeters is also [tex]$3$[/tex].
To solve this problem, we need to understand the relationship between the areas and the corresponding sides of similar figures. For similar figures, the ratio of their areas is equal to the square of the ratio of their corresponding sides. This is because area is a two-dimensional measurement, so it scales with the square of the linear dimensions.
Given that the area of one pentagon is 9 times that of the other, we can denote the ratio of the areas as [tex]$A_1 / A_2 = 9$[/tex]. If we let [tex]$k$[/tex] represent the ratio of the lengths of the corresponding sides, then we have [tex]$k^2 = 9$[/tex]. Taking the square root of both sides gives us [tex]$k = \sqrt{9}$[/tex], which simplifies to [tex]$k = 3$[/tex]. Therefore, the ratio of the lengths of the corresponding sides is [tex]$3$[/tex].
Now, since the pentagons are similar, the ratio of their perimeters will be the same as the ratio of their corresponding sides. This is because the perimeter is a one-dimensional measurement, and each side of the pentagon will be in the same ratio as the corresponding sides. Therefore, the ratio of the perimeters is also [tex]$3$[/tex].
In conclusion, the ratio of the lengths of the corresponding sides is [tex]$3$[/tex], and the ratio of the perimeters is also [tex]$3$[/tex]
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Answer:
The ratios of the lengths of the corresponding sides of the two pentagons is 3:1
The ratios of the perimeters of the two pentagons is 3:1
Step-by-step explanation:
Let the first pentagon be X and the second pentagon be Y.
Ratio of area of the two pentagons is [tex]\frac{A1}{A2} = \frac{9}{1}[/tex]
Ratio of sides of the pentagon is equal to
[tex]\frac{S1}{S2} = \sqrt{\frac{A1}{A2} }\\=\sqrt{\frac{9}{1} } \\=\frac{3}{1}[/tex]
Ratio of perimeters is equal to ration of sides
Hence,
[tex]\frac{P1}{P2} = \frac{3}{1}[/tex]
