A section of a tessellated plane is shown. Which type of symmetry does the tessellated plane have?

Mathematics

2 answers:

Ipatiy [6.2K]3 years ago

7 0

Answer:

This is a rotational symmetry.

Step-by-step explanation:

A tessellated plane is a flat surface made by tilting of the plane using tiles. These tiles do not overlap neither they have gaps.

There is rotational symmetry in the given pattern. While rotating the pattern, at a certain angle, we get the same exact pattern, this is why it is a rotational symmetry.

enot [183]3 years ago

3 0

It has rotational symmetry of order 3 about the centroid of any triangle. Turning the pattern 120° gives you the same pattern.

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2a: (c)

5o: (1, 3) and (1,1)

3a: (b)

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Step-by-step explanation:

2a: the equation of a circle circumference needs to be transformable to the form $(x-x_C)^2 + (y - y_C)^2 = r^2$ where $C(x_C; y_C)$ is the center and r is the radius. (a) and (d) can’t be it because they contain non-zero factors on xy. (b) isn’t an equation.

5o: just put the given (x, y) into the equations and see if it holds. (2, 3) isn’t on the circumference of (1) because $2^2+3^2-2\cdot 2 -4\cdot 3 + 4 = 1 \neq 0$ , (3, 1) isn’t on it either because $3^2 + 1^2 - 2\cdot 3 - 4\cdot 1 + 4 = 4 \neq 0$ .

3a: calculate the value of the left-hand side term of the equation using (x, y) from the given point M. That’s the difference of square distance to the center to the square radius $r^2$ . Thus it’s 0 if the point is on the circumference, negative if inside and positive if outside. You get $3^2 + 4^2 - 6\cdot 3 -2\cdot 4 + 8 = 7$ , positive, so it’s outside the circle.

1a: see definition from 2a. Here, $r^2 = 9$ .

4o: insert the y from the straight line equation (r) (which can be equivalently transformed to $y = x+1$ ) into the circumference equation. If it yields no solution, that’s outside, it there’s exactly one solution, that’s a tangent and if there are two solutions, it’s a secant. $x^2+(x+1)^2+2x - 4(x+1) = x^2 + x^2 + 2x + 1 + 2x - 4x + 4 = 2x^2 + 5 = 0 \iff x^2 = 2.5 \iff x \in \{-\sqrt{2.5}, \sqrt{2.5}\}$ There are two solutions, so it’s a secant.