Sam colors each tile in a 4 by 4 grid white or black. A coloring is called rotationally
1 answer:
Answer:
65,280
Step-by-step explanation:
Consider the 4×4 grid ...
where each of a, b, c, d is a 2×2 array of tiles. Let's use the notation a' to represent the 2×2 array "a" rotated right 1/4 turn. For 90° rotational symmetry, we must have b=a', c=b'=a'', d=c'=b''=a'''. That is, once "a" is determined, the rest of the grid is determined. Since "a" consists of 4 tiles, each of which can be black or white, there are 2^4 = 16 patterns that have 90° rotational symmetry.
The same will be true of 270° rotational symmetry, for the same reason.
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For 180° rotational symmetry, we must have c=a'' and d=b''. Then the combination of "a" and "b" together fully determines the grid. Together, "a" and "b" consist of 8 tiles, so there are 2^8 = 256 ways to pattern the grid so it will have 180° rotational symmetry. (Of those, 16 have 90° symmetry, and 16 have 270° symmetry. The sets are overlapping.)
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The 16 tiles of the grid can be colored 2^16 = 65,536 different ways. As we have seen, 256 of those colorings result in 180° rotational symmetry. Then the number of colorings that have no rotational symmetry is ...
65,536 -256 = 65,280 . . . . colorings not rotationally symmetric
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Answer: Angle A = 60°, angle B = 120°, angle C = 60°, angle D = 120°
Step-by-step explanation: Please refer to the attached diagram for further details.
The question has given the parallelogram ABCD as having side AB measuring 6 units, side AK measuring 3 units and line BK has been drawn perpendicular to line AD. This means line AK forms a right angle at the point where it meets line AD. From the information provided and the figure derived from that, we have right angled triangle ABK with the right angle at point K, line AB equals 6 units and line AK equals 3 units.
Using angle A as the reference angle, line AK is the opposite (side facing the reference angle), line AB is the hypotenuse (line facing the right angle) and line AK is the adjacent (side that lies between the right angle and the reference angle).
Hence, using the trigonometric ratios,
Cos A = adjacent/hypotenuse
Cos A = 3/6
Cos A = 0.5
Checking with the calculator, (that is second function of Cos 0.5)
A = 60
The opposite sides of a parallelogram are equal (that is line AD is parallel to line BC) and therefore opposite angles are equal (that is angle A is equal to angle C).
Similarly, angle B is equal to angle D. However, angles in a quadrilateral equals 360 degrees.
Therefore, angles in parallelogram ABCD is derived as;
A + B + C + D = 360
60 + B + 60 + D = 360
120 + B + D = 360
Subtract 120 from both sides of the equation
B + D = 240
Having known that angle B equals angle D, angles B and D is derived as each half of the value 240
Angle B = 240/2
Angle B = 120
and angle D = 120
Therefore, the angles are;
A = 60, B = 120, C = 60, D = 120
A graph which includes the possible values for number of people who can still sign up for the team is: B. number line with closed circle on 5 and shading to the right.
<h3>What is a number line?</h3>A number line can be defined as a type of graph with a graduated straight line which contains numerical values (positive and negative numbers) that are placed at equal intervals along its length.
Let the number of people who can still sign up for the team be represented by x. Thus, the inequality is given by:
x + 4 ≥ 9
x ≥ 9 - 4
x ≥ 5.
This ultimately implies that, there could be five (5) or more people that can still sign up for the team and a graph which includes these possible values is a number line with closed circle on five (5) and shading to the right because it can get larger.
Read more on number line here: brainly.com/question/24644930
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