Snowflakes have six identical arms. Using the centre of the snowflake as the point of rotation,what is the smallest positive angle through which you could rotate the snowflake so the arms line up?(The rotated snowflake would match the appearance of its original position.) Explain your answer. Hint: How far apart are the tips of the arms?
Please help I don't understand this!
You can use a hexagon instead of a snowflake.
Think of a straight line. If you stick a pin in the center, and turn it 180°, it looks just as it did before turning it.
Take an equilateral triangle. Stick a pin in the center and turn it. After 120°, the bottom corner is where the top corner started out, and the triangle looks unchanged.
Take a square. After turning it 90°, it looks just as it did when you started.
So, for a regular polygon of n sides, if you turn it 360/n°, it looks unchanged.
So, for a snowflake with 6-fold symmetry, if you stick a pin in the center and start turning it, after 360/6 = 60°, it looks the way it started.
To find the smallest positive angle through which you can rotate the snowflake so the arms line up, we need to consider the arrangement of the arms and the distance between their tips.
First, let's imagine that we have a complete rotation of 360 degrees. Since the snowflake has six arms, each rotation of 360 degrees would bring the arms back to their original positions, lining up perfectly. However, we are looking for the smallest positive angle, which means we want to rotate the snowflake as little as possible while still achieving the desired alignment.
Since the snowflake has six arms, we can divide 360 degrees by 6 to find the angle between each neighboring arm. Thus, 360 degrees divided by 6 equals 60 degrees. This means that each arm is 60 degrees apart from its neighboring arm.
To line up the arms, we need to find the distance between the tips of two neighboring arms. Since the arms are identical and symmetrically arranged, their tips are equidistant from the center of the snowflake.
Now, let's draw an equilateral triangle with its vertices at the center of the snowflake and the tips of two neighboring arms. An equilateral triangle has all sides and angles equal. Since the tips of two neighboring arms are equidistant from the center, the sides of the equilateral triangle are also equal.
Let's call the distance between the tips of two neighboring arms "d." Now, we have a triangle with three sides of length "d" and three angles of 60 degrees each.
Using trigonometry, we can calculate the length of one side of the equilateral triangle. By dividing the triangle into two right-angled triangles, we can use the trigonometric relationship for right-angled triangles.
In a right-angled triangle, the length of one side (adjacent) divided by the length of another side (hypotenuse) is equal to the cosine of the angle between them.
In our equilateral triangle, two sides of equal length "d" form a right angle, and one side is the hypotenuse. Since the angle between these two sides is 60 degrees, we can write:
cos(60 degrees) = d / d
Simplifying the equation, we get:
1/2 = d / d
Therefore, d = (1/2) * d
This implies that d = 1/2.
Hence, the distance between the tips of two neighboring arms is 1/2, which means that the smallest positive angle through which you can rotate the snowflake so the arms line up is 60 degrees.