Two rectangles are similar the ratio of the lengths of their corresponding sides is 1:2 find the ratio of the peremeters of the two rectangles the find the ratio of the areas explain your answers
a. P1/P2 =(2L+2W)/(4L+4W)=
2(L+W)/4(L+W) = 2/4 = 1/2.
b. A1/A2 = L*W/(2L*2W)=(L*W)/4(L*W=1/4.
(L+W=2/4
= 1/2.
To find the ratio of the perimeters, we'll first consider the ratio of the lengths of the corresponding sides. Let's call this ratio "k". Given that the ratio of the lengths of the sides is 1:2, we can say k = 1/2.
Let's assume the lengths of the sides of the first rectangle are a and b. Consequently, the lengths of the sides of the second rectangle will be 2a and 2b, as they are twice the size.
The perimeter of the first rectangle is given by the sum of all four sides, which is 2a + 2b = 2(a + b).
Similarly, the perimeter of the second rectangle will be 2(2a + 2b) = 4(a + b). Thus, the ratio of the perimeters of the two rectangles is:
(2(a + b)) : (4(a + b))
We can simplify this ratio by dividing both terms by 2:
(a + b) : 2(a + b)
The ratio of the perimeters is 1:2.
Next, let's find the ratio of the areas of the two rectangles.
The area of the first rectangle is given by a * b, and the area of the second rectangle is 2a * 2b.
The ratio of the areas is:
(a * b) : (2a * 2b)
Simplifying this ratio, we get:
(a * b) : (4ab)
Now, we divide both terms by ab:
a : 4a
This simplifies further to:
1 : 4
Thus, the ratio of the areas of the two rectangles is 1:4.
To find the ratio of the perimeters of the two rectangles, let's first assign some variables.
Let the lengths of the corresponding sides of the smaller rectangle be x and y, and the lengths of the corresponding sides of the larger rectangle be 2x and 2y.
The ratio of the perimeters is the sum of all corresponding sides of the larger rectangle divided by the sum of all corresponding sides of the smaller rectangle.
The perimeter of the larger rectangle is 2(2x + 2y) = 4x + 4y.
The perimeter of the smaller rectangle is 2(x + y) = 2x + 2y.
The ratio of the perimeters is (4x + 4y) / (2x + 2y). We can simplify this ratio by cancelling out the common factors of 2:
(4x + 4y) / (2x + 2y) = (2(2x + 2y)) / (2(x + y))
= 2x + 2y / x + y
So, the ratio of the perimeters of the two rectangles is 2x + 2y / x + y.
Now, let's find the ratio of the areas of the two rectangles. The area of a rectangle is given by the length multiplied by the width.
The area of the larger rectangle is (2x)(2y) = 4xy.
The area of the smaller rectangle is (x)(y) = xy.
The ratio of the areas is 4xy / xy = 4.
So, the ratio of the areas of the two rectangles is 4.
To summarize:
- The ratio of the perimeters of the two rectangles is 2x + 2y / x + y.
- The ratio of the areas of the two rectangles is 4.