If the ratio of the areas of two similar hexagons is 4:49, find the ratio of their apothems
for similar figures areas scale with the square of lengths
so
sqrt(4/49) = 2/7
To find the ratio of the apothems of two similar hexagons, we can make use of the fact that the ratio of the areas of two similar polygons is the square of the ratio of their corresponding side lengths.
Let's denote the ratio of the areas as 4:49, which means the first hexagon has an area of 4x and the second hexagon has an area of 49x (where x is a constant).
The area of a regular hexagon is given by the formula: A = (3√3/2) * s^2, where A is the area and s is the length of a side.
Let's denote the side length of the first hexagon as s1 and the side length of the second hexagon as s2. We can set up the following equation using the given ratio of areas: (3√3/2) * s1^2 = 4x and (3√3/2) * s2^2 = 49x.
Dividing the second equation by the first, we get:
s2^2 / s1^2 = (49x) / (4x).
Simplifying further:
(s2 / s1)^2 = 49 / 4.
Taking the square root of both sides to find the ratio of the side lengths:
s2 / s1 = √(49 / 4).
Since we are looking for the ratio of the apothems, we know that the apothem of a regular hexagon is equal to (s * √3) / 2.
Let's denote the apothem of the first hexagon as a1 and the apothem of the second hexagon as a2. We can use the ratio of the side lengths to find the ratio of the apothems:
a2 / a1 = (s2 * √3) / 2 / (s1 * √3) / 2.
Canceling out the √3 and simplifying:
a2 / a1 = s2 / s1.
Therefore, the ratio of the apothems of the two similar hexagons is equal to the ratio of their side lengths, which is √(49 / 4) or 7/2.