One of two similar triangles has an area of 1/4 times than that of the other. What is the ration of the trangles' perimeter?
(I know that the answer is 1:2, but i don't why that is the answer :( )
To determine the ratio of the perimeters of two similar triangles, we need to understand how the areas of similar triangles are related to their side lengths.
First, let's establish a relationship between the areas of two similar triangles. Recall that the area of a triangle is given by A = (1/2) * base * height. Since we're working with similar triangles, their corresponding sides are proportional.
Let's assume that the two triangles have side lengths a, b, and c for the first triangle, and x, y, and z for the second triangle, respectively.
Given that the area of the first triangle is 1/4 that of the second triangle, we can write the following equation:
(1/4) * (1/2) * a * b = (1/2) * x * y * z
Simplifying the equation, we have:
(1/8) * a * b = (1/2) * x * y * z
Next, let's compare the perimeters of the two triangles. The perimeter of a triangle is the sum of its three sides.
The perimeter of the first triangle is a + b + c, and the perimeter of the second triangle is x + y + z.
Since the triangles are similar, we know that the ratios of the corresponding sides are the same, meaning:
a : x = b : y = c : z
We can express this relationship as:
a/x = b/y = c/z = k
Now, let's find the ratio between the perimeters:
(a + b + c) / (x + y + z) = (a/x + b/y + c/z)
Using the ratio relationship we found, we can substitute the values:
(a/x + b/y + c/z) = (k + k + k) = 3k
Therefore, the ratio of the perimeters of the two triangles is 3k.
Now, taking into account the area relationship we established earlier, where the area of the first triangle is 1/4 that of the second triangle, we can substitute this relationship into the perimeter ratio equation:
3k = 1/4
Simplifying, we find that k = 1/12.
Substituting this value back into the perimeter ratio equation, we have:
3k = 3 * (1/12) = 1/4
Therefore, the ratio of the perimeters is 1:4.
I apologize for the confusion in my initial response. The correct ratio is 1:4, not 1:2.