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Answer :
* The perimeter of the larger pentagon is expressed in terms of $x$ as $P_L = 2x + 14$.
* Using the constant of proportionality, the perimeter of the larger pentagon is calculated as $P_L = 32$.
* Solving for $x$, we find $x = 9$.
* The longest side of the larger pentagon is $\boxed{9}$.
### Explanation
1. Problem Analysis
We are given two similar pentagons. The side lengths of the larger pentagon are $8, x-3, 4, 5,$ and $x$. The perimeter of the smaller pentagon is 24 units, and the constant of proportionality (ratio of smaller to larger pentagon) is $\frac{3}{4}$. Our goal is to find the length of the longest side of the larger pentagon.
2. Perimeter of Larger Pentagon
Let $P_L$ be the perimeter of the larger pentagon. Then, we can express $P_L$ as the sum of its side lengths: $$P_L = 8 + (x-3) + 4 + 5 + x = 2x + 14.$$
3. Ratio of Perimeters
Since the pentagons are similar, the ratio of their perimeters is equal to the constant of proportionality. Therefore, we have $$\frac{24}{P_L} = \frac{3}{4}.$$
4. Solving for the Perimeter
Now, we solve for $P_L$: $$P_L = \frac{4 \cdot 24}{3} = \frac{96}{3} = 32.$$
5. Substituting the Perimeter
Substitute $P_L = 32$ into the equation $P_L = 2x + 14$: $$32 = 2x + 14.$$
6. Solving for x
Solve for $x$: $$2x = 32 - 14 = 18,$$ $$x = \frac{18}{2} = 9.$$
7. Finding Side Lengths
Now we find the side lengths of the larger pentagon by substituting $x = 9$: The side lengths are $8, x-3, 4, 5, x$, which are $8, 9-3, 4, 5, 9$, or $8, 6, 4, 5, 9$.
8. Finding the Longest Side
The longest side of the larger pentagon is 9 units.
### Examples
Understanding similarity and proportionality is crucial in various real-world applications, such as scaling architectural designs, creating accurate maps, and even in art, where artists use proportions to create realistic drawings and sculptures. For instance, when an architect creates a blueprint of a building, they use similar figures to represent the actual building. The constant of proportionality ensures that all dimensions are scaled correctly, maintaining the building's structural integrity and aesthetic appeal.
* Using the constant of proportionality, the perimeter of the larger pentagon is calculated as $P_L = 32$.
* Solving for $x$, we find $x = 9$.
* The longest side of the larger pentagon is $\boxed{9}$.
### Explanation
1. Problem Analysis
We are given two similar pentagons. The side lengths of the larger pentagon are $8, x-3, 4, 5,$ and $x$. The perimeter of the smaller pentagon is 24 units, and the constant of proportionality (ratio of smaller to larger pentagon) is $\frac{3}{4}$. Our goal is to find the length of the longest side of the larger pentagon.
2. Perimeter of Larger Pentagon
Let $P_L$ be the perimeter of the larger pentagon. Then, we can express $P_L$ as the sum of its side lengths: $$P_L = 8 + (x-3) + 4 + 5 + x = 2x + 14.$$
3. Ratio of Perimeters
Since the pentagons are similar, the ratio of their perimeters is equal to the constant of proportionality. Therefore, we have $$\frac{24}{P_L} = \frac{3}{4}.$$
4. Solving for the Perimeter
Now, we solve for $P_L$: $$P_L = \frac{4 \cdot 24}{3} = \frac{96}{3} = 32.$$
5. Substituting the Perimeter
Substitute $P_L = 32$ into the equation $P_L = 2x + 14$: $$32 = 2x + 14.$$
6. Solving for x
Solve for $x$: $$2x = 32 - 14 = 18,$$ $$x = \frac{18}{2} = 9.$$
7. Finding Side Lengths
Now we find the side lengths of the larger pentagon by substituting $x = 9$: The side lengths are $8, x-3, 4, 5, x$, which are $8, 9-3, 4, 5, 9$, or $8, 6, 4, 5, 9$.
8. Finding the Longest Side
The longest side of the larger pentagon is 9 units.
### Examples
Understanding similarity and proportionality is crucial in various real-world applications, such as scaling architectural designs, creating accurate maps, and even in art, where artists use proportions to create realistic drawings and sculptures. For instance, when an architect creates a blueprint of a building, they use similar figures to represent the actual building. The constant of proportionality ensures that all dimensions are scaled correctly, maintaining the building's structural integrity and aesthetic appeal.
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