One of two similar triangles has an area of 1/4 times that of the other. What is the ratio of its perimeter of the triangles?
(I know the answer but I don't know the solution, please explain. Thank you very much!!)
based on the theorem that the area of similar figures is proportional to the square of their corresponding sides, thus also their perimeters.
so
(p1)^2 : (p2)^2 = 1 : 4
p1 : p2 = √1 : √4
= 1 : 2
To find the ratio of the perimeters of two similar triangles, we first need to understand the relationship between their areas and side lengths.
Let's denote the area of the first triangle as A1, and the area of the second triangle as A2. Accordingly, let the perimeters of the triangles be P1 and P2.
We are given that the area of one triangle is 1/4 times that of the other:
A2 = 1/4 * A1
Now, we know that the area of a triangle is given by the formula:
Area = (1/2) * base * height
Since the given triangles are similar, the ratio of their areas will be equal to the square of the ratio of their corresponding side lengths. Let's denote the ratio of their side lengths as k:
A2/A1 = (k²)^2 = k^4
Substituting the given relationship between the areas, we get:
1/4 * A1/A1 = k^4
1/4 = k^4
Next, let's consider the relationship between perimeters and side lengths. The perimeter of a triangle is simply the sum of its three sides.
Since the triangles are similar, the ratio of their corresponding side lengths will be the same as the ratio of their perimeters:
P2/P1 = k
Now we have two equations:
1/4 = k^4 Equation (1)
P2/P1 = k Equation (2)
To solve for the ratio of the perimeters, we need to find the value of k. We can take the fourth root of both sides of Equation (1):
(k^4)^(1/4) = (1/4)^(1/4)
k = 1/√2
Substituting this value of k into Equation (2), we get:
P2/P1 = 1/√2
Hence, the ratio of the perimeters of the two similar triangles is 1/√2.
To solve this problem, we can start by understanding the relationship between the areas of two similar triangles.
When two triangles are similar, their corresponding sides are proportional in length. This means that if the ratio of their corresponding side lengths is a:b, then the ratio of their areas will be equal to the square of this ratio, which is (a/b)^2.
Let's assume that the two similar triangles have corresponding side lengths in the ratio of a:b. Therefore, the ratio of their areas would be (a/b)^2.
In this problem, we are given that one triangle has an area that is 1/4 times that of the other. So, we can set up the equation:
(a/b)^2 = 1/4
To solve for the ratio of their perimeters, we need to understand the relationship between the perimeters of similar triangles. Since the sides of the triangles are proportional, the ratio of their perimeters will be the same as the ratio of their corresponding side lengths. Hence, the ratio of the perimeters will also be a:b.
We can rewrite this ratio as a/b = √(1/4) = 1/2.
Therefore, the ratio of the perimeter of the two similar triangles is 1:2.
In summary:
- The ratio of the areas of similar triangles is the square of the ratio of their corresponding side lengths.
- The ratio of the perimeters of similar triangles is equal to the ratio of their corresponding side lengths.
By using these relationships and given information of the area ratio of 1:4, we can determine that the ratio of the perimeter is 1:2.