In a regular hexagon, what is the ratio of the length of the shortest diagonal to the length of the longest diagonal? Express your answer as a common fraction in the simplest radical form.
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To work on its "diagonal" physique!
In a regular hexagon, any diagonal that connects non-adjacent vertices is called a longest diagonal and any diagonal that connects adjacent vertices is called a shortest diagonal.
Let's name the length of a side of the hexagon as "s".
The length of the shortest diagonal can be found by drawing a line from any vertex to the vertex adjacent to it. This line will divide the hexagon into two congruent triangles. Each of these triangles will have a base equal to "s" and a height equal to half the length of the longest diagonal (which is also equal to the length of a side of the hexagon).
Using the Pythagorean theorem, we can find the length of the longest diagonal.
The Pythagorean theorem states that the square of the length of the hypotenuse of a right triangle is equal to the sum of the squares of the lengths of the other two sides.
Applying the Pythagorean theorem to one of the congruent triangles, we have:
(longest diagonal)^2 = (base)^2 + (height)^2
(longest diagonal)^2 = s^2 + (s/2)^2
(longest diagonal)^2 = s^2 + s^2/4
(longest diagonal)^2 = (4s^2 + s^2)/4
(longest diagonal)^2 = (5s^2)/4
(longest diagonal) = sqrt(5s^2)/2
(longest diagonal) = s * sqrt(5)/2
The ratio of the length of the shortest diagonal to the length of the longest diagonal is:
(shortest diagonal) / (longest diagonal) = s / (s * sqrt(5)/2)
(shortest diagonal) / (longest diagonal) = 1 / (sqrt(5)/2)
(shortest diagonal) / (longest diagonal) = 2 / sqrt(5)
Therefore, the ratio of the length of the shortest diagonal to the length of the longest diagonal is 2/sqrt(5).
To find the ratio of the shortest diagonal to the longest diagonal in a regular hexagon, we need to determine the lengths of these diagonals.
In a regular hexagon, all sides are congruent and all angles are equal. Each diagonal that connects non-adjacent vertices divides the hexagon into two congruent triangles.
Let's consider one of these triangles. We can draw the three diagonals that connect the vertices of the triangle. One of these diagonals is the side of the triangle, and the other two are the diagonals of the hexagon.
The shortest diagonal in a regular hexagon is equal to the side length of the hexagon. Let's call this length "s".
To find the length of the longest diagonal, we need to use the Pythagorean theorem. The two diagonals that connect the vertices of the triangle create a right triangle. One leg of this triangle is equal to the side length "s", and the other leg is equal to half the side length, or "s/2". The longest diagonal, which is the hypotenuse, can be found using the Pythagorean theorem:
(longest diagonal)^2 = (side length)^2 + (half side length)^2
d^2 = s^2 + (s/2)^2
d^2 = s^2 + s^2/4
d^2 = (4s^2 + s^2)/4
d^2 = 5s^2/4
Taking the square root of both sides, we have:
d = sqrt(5s^2)/2
Now we can find the ratio of the shortest diagonal (s) to the longest diagonal (d):
s/d = s / (sqrt(5s^2)/2)
To simplify, we can multiply the numerator and denominator by 2/sqrt(5):
s/d = (s * 2 / sqrt(5s^2)) * (2 / sqrt(5s^2))
s/d = 4s / sqrt(5s^2)
s/d = (4/5) * sqrt(5s^2)
Therefore, the ratio of the length of the shortest diagonal to the length of the longest diagonal in a regular hexagon is (4/5) * sqrt(5s^2), where "s" represents the side length of the hexagon.
To find the ratio of the lengths of the shortest diagonal to the longest diagonal in a regular hexagon, let's consider the diagonals that connect opposite vertices.
In a regular hexagon, all sides and angles are equal. Let's label the length of one side as "s".
Now, if we draw the diagonals that connect opposite vertices of the hexagon, we can divide the hexagon into six congruent equilateral triangles.
One of these equilateral triangles will have side length "s", and since all sides are equal, all the equilateral triangles will have side length "s".
The diagonals of each equilateral triangle will bisect the angles, forming two congruent right triangles.
Let's focus on one of these right triangles. The length of the shorter leg (the half of the shortest diagonal) can be found by drawing a perpendicular from a vertex to the opposite side. It divides the equilateral triangle into two congruent 30-60-90 right triangles.
In a 30-60-90 triangle, the ratio of the sides is always 1 : √3 : 2. In our case, the shorter leg (half of the shortest diagonal) is represented by 1.
So, the length of the shorter leg (half of the shortest diagonal) is "s/2".
Similarly, the longer leg (half of the longest diagonal) is "s√3".
Now, let's find the ratio of the shortest diagonal to the longest diagonal.
The shortest diagonal is formed by the shorter leg of the right triangle, which is "s/2".
The longest diagonal is formed by the longer leg of the right triangle, which is "s√3".
Therefore, the ratio is:
(s/2) / (s√3) = 1 / (2√3) = 1 / (2√3) * (√3/√3) = √3 / (6) = √3 / 6.
So, the ratio of the length of the shortest diagonal to the length of the longest diagonal in a regular hexagon is √3 / 6.
Hint: Pick any apex of the hexagon, point A. Let B be one of the 2 apices such that AB is a short diagonal. Let C be the apex such that AC is a long diagonal. BC will be a side of the hexagon.
Triangle ABC will be a right angle triangle. Long diagonal AC will be the hypotenuse. Use Pythagorus' rule to relate the lengths.
The interior angle of a regular hexagon is 120°. Use the law of cosines and regularity of the hexagon to express the length of the short diagonal to the side of the hexagon.
Solve these simultaneous equations.