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Mastering Rotational Symmetry and Transformations in Geometry

TX.G.10.A

Surface area of 3-dimensional shapes

Surface area and volume of prisms

Surface area and volume of pyramids

Surface area and volume of cylinders

Surface area and volume of cones

Surface area and volume of spheres

TX.G.10.B

auto-gen

Upper and lower bound

Upper and Lower Bound

TX.G.11.B

Areas and sectors of circles

TX.G.11.C

Introduction to surface area of 3-dimensional shapes

Nets of 3-dimensional shapes

TX.G.11.D

Introduction to volume

Calculate the Volume

TX.G.12.A

Angles in a circle

Chord properties

Tangent properties

Circles and circumference

Circle properties

Central angles and proofs

Inscribed angles and proofs

Central and inscribed angles in circles

TX.G.12.B

Arcs of a circle

Circle chord, tangent, and inscribed angles proofs

TX.G.12.D

Converting between degrees and radians

Trigonometric ratios of angles in radians

Radian measure and arc length

TX.G.12.E

Conics - Circle

TX.G.13.A

Permutations

Combinations

Problems involving both permutations and combinations

TX.G.13.B

Probability

TX.G.13.C

Probability of independent events

Addition rule for 

Addition Rule for "OR"

Addition rule for "OR"

Multiplication rule for 

Multiplication rule for "AND"

Multiplication Rule for "AND"

TX.G.13.D

Conditional probability

Probability with Venn diagrams

TX.G.13.E

Determining probabilities using tree diagrams and tables

Probability involving permutations and combinations

TX.G.2.A

Midpoint formula: \(M = ( \frac{x_1+x_2}2  ,\frac{y_1+y_2}2)\)

Midpoint formula

Midpoint formula: <katex>M = ( \frac{x_1+x_2}2  ,\frac{y_1+y_2}2)</katex>

Midpoint formula: <katex>M = ( \frac{x_1+x_2}2 ,\frac{y_1+y_2}2)</katex>

TX.G.2.B

Distance formula: \(d = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\)

Distance formula: <katex>d = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}</katex>

Distance formula

Slope equation: \(m = \frac{y_2-y_1}{x_2- x_1}\)

Slope equation: <katex>m = \frac{y_2-y_1}{x_2- x_1}</katex>

Parallel and perpendicular lines in linear functions

TX.G.2.C

Parallel line equation

Perpendicular line equation

Combination of both parallel and perpendicular line equations

TX.G.3.A

Rotational symmetry and transformations

Transformations of functions: Horizontal translations

Transformations of functions: Vertical translations

TX.G.3.B

Combining transformations of functions

Enlargements and reductions with scale factors

TX.G.3.C

Scale diagrams

TX.G.3.D

Line symmetry

Reflection across the y-axis: y = f(-x)

Reflection across the x-axis: y = -f(x)

TX.G.4.A

Inductive reasoning

Deductive reasoning

TX.G.4.B

Conditionals

Inverses, converses, and contrapositives

Biconditionals

TX.G.4.D

Algebraic proofs

TX.G.5.A

Congruence and congruent triangles

Pairs of lines and angles

Parallel lines and transversals

Perpendicular line proofs

TX.G.5.D

Classifying triangles

TX.G.6.A

Isosceles and equilateral triangles

Parallel line proofs

TX.G.6.B

Triangles congruent by SSS proofs

Triangles congruent by SAS and HL proofs

Triangles congruent by ASA and AAS proofs

TX.G.6.D

Combination of SohCahToa questions

Pythagorean theorem

Pythagorean Theorem

Using the pythagorean relationship

Applications of pythagorean theorem

TX.G.6.E

Classifying quadrilaterals

Classifying Quadrilaterals

Word Problem: Classifying Quadrilaterals

TX.G.7.A

Similar triangles

Similar polygons

TX.G.8.B

Estimating square roots

TX.G.9.A

Use sine ratio to calculate angles and sides (Sin = o / h)

Use cosine ratio to calculate angles and sides (Cos = a / h)

Use tangent ratio to calculate angles and sides (Tan = o / a)

TX.G.9.B

Solving expressions using 45-45-90 special right triangles

Solving expressions using 30-60-90 special right triangles

Circles

Coordinate and transformational geometry

Logical argument and constructions

Proof and congruence

Two-dimensional and three-dimensional figures

math

Intro

Ex. 1:

Ex. 2:

<p>- Hi, welcome to this question right here. So here we are. This figure right here, it says in the question it does have a rotation symmetry. So the first part of the question asks you, "Where is the center of rotation?" Now, all you have to do is just pick the center of your image and they have to know that this point, if you rotate it, it will form exactly the same image after you rotate it to certain degrees. So for this figure right here is a rectangular shape, the green part, right? So the center of the symmetry, or I'd say the center of rotation will be this point right here. So you can imagine that if you have a piece of paper with this size and if you rotate that piece of paper around, you will still get the same image, okay? So what are the order and the angle of rotation? The order of the rotation is this. You can actually relate the order to the line of symmetry you have. So if you look at the square right here, you have four symmetrical lines. You got one this way, this way, and this way, and then this way. So there's four lines. So technically the order of rotation for this question will be a four, okay? Now, the angle of rotation, you can actually calculate it. It's pretty simple. You see, when we rotate things around, it's actually 360. So to calculate this, it's 360 and all you have to do is divide it by the order number that you found, which is four. Now once again, to figure the order, all you have to do is figure a line of symmetry you have. So let's just quickly draw that. So we&apos;ve got one line going down this way. This is one symmetrical line and this is two and this is three and this is four. So with four lines of symmetry, you can actually relate that to the order number. So to calculate the angle of rotation, just take the 360, which is one complete turn, it's always 360 by the way, and divide it by four, the orders that you have. Now 360 divided by four, using your calculator, you get yourself a 90 degrees. So that's your angle of rotation right there, 90 degrees. It says, "Show your answers in degrees," which is this one right here and/or as a fraction of a turn. Now, a fraction of a turn, it's 360, right? So if you take that and divide it by 360, which is 90 divided by 360 again, basically you get yourself a one quarter, and this is your one quarter of a turn for this question. That's it. Thanks for watching.</p>


When an object is turned around its center of rotation to certain <a href='[[glossary.html|{"keyword":"Degree","term":"Degree"}]]'>degrees</a> and the object looks the same, we can say it has rotational symmetry. We will learn the details of rotational symmetry including, lines of symmetry, order of rotation, and angle of rotation.   