The ratio of the areas of two similar triangles is 36 : 25. What is the ratio of the perimeters of the two triangles?
To find the ratio of the perimeters of two similar triangles, we can use the fact that the ratio of corresponding sides of similar triangles is the same as the ratio of their perimeters. We are given the ratio of the areas, so we need to find the ratio of the sides first.
Since the ratio of the areas of the two triangles is 36 : 25, we know that the ratio of their corresponding side lengths squared is also 36 : 25. To find the ratio of their side lengths, we take the square root of each number:
√(36/25) = 6/5
So, the ratio of the side lengths of the two triangles is 6 : 5.
Now, we can use this ratio to find the ratio of their perimeters. Since the perimeter is the sum of all the sides, the ratio of their perimeters will also be 6 : 5.
Therefore, the ratio of the perimeters of the two triangles is 6 : 5.
To find the ratio of the perimeters of the two triangles, we need to know that the ratio of their corresponding sides is the same as the ratio of their areas. In other words, if the ratio of the areas of two similar triangles is A:B, then the ratio of their corresponding sides is √(A:B).
In this case, the ratio of the areas is given as 36:25. Since the areas are proportional to the squares of the corresponding sides, we can find the ratio of the sides by taking the square root of 36:25.
√(36:25) = √(6²:5²) = 6:5
Therefore, the ratio of the perimeters of the two triangles is also 6:5.
If the similarity ratio of two similar triangles is a : b, then the ratio of their areas is :
a ^ 2 / b ^ 2
So :
a ^ 2 / b ^ 2 = 36 / 25
a / b = sqrt ( 36 ) / sqrt ( 25 ) = 6 / 5