The perimeters of 2 similar polygons are 12 and 16.
1. What is the ratio of the lengths of 2 corresponding sides?
2. What is the ratio of the areas?
3. If the area of the smaller polygon is 54 what is the area of the larger polygon?
ratio of ANY lengths in similar figures is the same 12/16 = 3/4
2. Ratios of ANY areas is length ratio squared 9/16
3. 54 * 16/9 = 96
1. Well, since the perimeters of the polygons are in a 3:4 ratio, we can say that the lengths of the corresponding sides are in a 3:4 ratio as well. But hey, don't judge the smaller polygon just because it's a little short on sides!
2. When it comes to the ratio of the areas, it's the square of the ratio of the corresponding sides. So, in this case, it would be (3/4)^2. Now that's some math that even the polygons can square up to!
3. If the area of the smaller polygon is 54, then all you need to do is multiply it by the square of the ratio of the corresponding sides. So, the area of the larger polygon would be 54*(4/3)^2. Now that's a bigger area to play around with!
Remember, numbers can be fun too! Just make sure to keep those polygons in shape, otherwise, they might start acting all odd-sided on us!
To find the answers to these questions, we can use the fact that similar polygons have corresponding sides that are proportional.
1. To find the ratio of the lengths of two corresponding sides, divide the perimeters of the polygons:
Ratio of lengths of corresponding sides = Perimeter of larger polygon / Perimeter of smaller polygon
Ratio of lengths of corresponding sides = 16 / 12
Ratio of lengths of corresponding sides = 4 / 3
2. The ratio of the areas of two similar polygons is equal to the square of the ratio of their corresponding sides.
Ratio of areas = (Ratio of lengths of corresponding sides)^2 = (4/3)^2 = 16/9
3. If the area of the smaller polygon is 54, we can use the ratio of areas to find the area of the larger polygon:
Area of larger polygon = Area of smaller polygon * Ratio of areas
Area of larger polygon = 54 * 16/9
Area of larger polygon = 96
To find the ratios and area of similar polygons, we need to use the properties of similar figures.
1. Ratio of the lengths of corresponding sides:
The ratio of the perimeters of two similar polygons is equal to the ratio of their corresponding side lengths. In this case, the ratio of perimeters is given as 12:16, which simplifies to 3:4. Therefore, the ratio of the lengths of the corresponding sides is 3:4.
2. Ratio of the areas:
The ratio of the areas of two similar polygons is equal to the square of the ratio of their corresponding side lengths. Based on our previous answer, the ratio of the lengths of the corresponding sides is 3:4. Squaring this ratio gives us 9:16. Therefore, the ratio of the areas is 9:16.
3. Finding the area of the larger polygon:
If the area of the smaller polygon is given, we can use the ratios of the areas to find the area of the larger polygon. In this case, the area of the smaller polygon is given as 54. We know that the ratio of the areas is 9:16, so we can set up the following proportion:
(9/16) = (54/L), where L represents the area of the larger polygon.
By cross-multiplying, we get:
9L = 54 * 16
Dividing both sides by 9:
L = (54 * 16) / 9
Evaluating the expression on the right side gives us:
L = 96
Therefore, the area of the larger polygon is 96.