Question

transformations test (topic 4)

let x and y be the following sets:
x = degrees of rotation of a octagon (8 - sided) up to 360
y = degrees of rotation of a square up to 360

what is the set for the following?
( x cup y )

options:

• (90, 180, 270, 360)
• {0}
• (45, 90, 135, 180, 225, 270, 315, 360)
• (45, 135, 225, 315)

Explanation:

Step1: Find set X (octagon rotations)

A regular octagon has 8 sides. The rotational symmetry angles (up to \(360^\circ\)) are found by \(\frac{360^\circ}{n}\times k\), where \(n = 8\) and \(k = 0,1,\dots,7\). So \(\frac{360}{8}=45^\circ\), so \(X=\{0, 45, 90, 135, 180, 225, 270, 315, 360\}\) (note: \(k = 0\) gives \(0^\circ\), \(k = 1\) gives \(45^\circ\), etc., up to \(k = 8\) which is \(360^\circ\), but usually included as a full rotation).

Step2: Find set Y (square rotations)

A square has 4 sides. Rotational symmetry angles: \(\frac{360^\circ}{4}\times k\), \(k = 0,1,2,3\). So \(\frac{360}{4}=90^\circ\), so \(Y = \{0, 90, 180, 270, 360\}\).

Step3: Find \(X\cup Y\) (union of sets)

The union of two sets contains all elements from either set. Combining \(X\) and \(Y\), we get elements: \(0, 45, 90, 135, 180, 225, 270, 315, 360\) (since \(X\) already includes the elements of \(Y\) plus \(45, 135, 225, 315\)). Looking at the options, the set \(\{45, 90, 135, 180, 225, 270, 315, 360\}\) (assuming \(0\) might be a typo or excluded in the option, but the third option matches the non - zero and included angles from the union) is the correct union.

Answer:

\(\{45, 90, 135, 180, 225, 270, 315, 360\}\) (the third option in the given choices)