The areas of two similar triangles are 64 and 100. what is the ratios of their corresponding altitudes?
square root of the area ratio
area scales as the square of the linear lengths. So, the altitudes are in the ratio √(64/100) = 8/10 = 4:5
To find the ratios of the corresponding altitudes of two similar triangles, we need to find the ratio of their corresponding areas first. Since the areas of the triangles are given, we can calculate the ratio of the corresponding areas as follows:
Ratio of areas = (Area of one triangle) / (Area of the other triangle)
Let's assume the areas of the two triangles are A1 and A2, and their corresponding altitudes are h1 and h2.
Given that the areas of the two triangles are 64 and 100, we can write:
A1 = 64
A2 = 100
To find the ratio of their areas, we can use the formula:
Ratio of areas = A1 / A2
Substituting the given values:
Ratio of areas = 64 / 100 = 0.64
So, the ratio of the areas is 0.64.
Now, since the triangles are similar, we know that the ratio of their corresponding side lengths is equal to the square root of the ratio of their areas. Therefore, the ratio of their corresponding altitudes is the same as the ratio of their corresponding side lengths.
In this case, the ratio of the corresponding side lengths (and altitudes) is the square root of 0.64.
Calculating the square root of 0.64:
√0.64 ≈ 0.8
Therefore, the ratio of the corresponding altitudes is approximately 0.8.