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Mastering Rotational Symmetry and Transformations in Geometry

Geometry: Master Shapes, Proofs & Spatial Reasoning

Measuring Systems

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Conversions between metric and imperial systems

Metric to Imperial Conversions

Imperial to Metric Conversions

Conversions Between Metric and Imperial Systems Speed KM to MI

Conversions Between Metric and Imperial Systems Speed  MI to M

Metric-Imperial Length Conversion Inch To Cm

Metric-Imperial Length Conversion MM/CM/M/KM To FT

Metric-Imperial Length Conversion MM/CM/M/KM To MI

Metric systems in measuring systems

Add and Subtract Mixed Units

Metric Length Conversion

Metric systems

Add or Subtract Different Metric Units Convert to CM

Add or Subtract Different Metric Units Convert to M

Add or Subtract Different Metric Units Convert to MM

Imperial systems in measuring systems

Imperial systems

Imperial Length Conversion

Conversions involves squares and cubes

Imperial to Metric Conversions — Squares and Cubes

Imperial Conversion — Squares and Cubes

Metric to Imperial Conversion — Squares and Cubes

Conversions involving squares and cubic

Metric Area Conversion FT to CM

Metric Area Conversion Foot to Inch

Metric Area Conversion M/CM to Inch

Surface Area and Volume

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

Points, Lines, Segments, Midpoints and Chords

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

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>

Slope-Y-intercept form: y = mx + b

Slope intercept form: y = mx + b

Graph using a single point and slope

Solve linear equations

Introduction to linear equations

Circles, plane and coordinate

Angles in a circle

Chord properties in circle geometry

Tangent properties in circle geometry

Circles and circumference 

Circle properties

Arcs of a circle

Areas and sectors of circles

Central angles and proofs

Inscribed angles and proofs

Central and inscribed angles in circles

Circle chord, tangent, and inscribed angles proofs

Triangles and special triangles

Solving expressions using 45-45-90 special right triangles

Solving expressions using 30-60-90 special right triangles

Pythagorean theorem

Pythagorean Theorem

Classifying triangles

Congruence and congruent triangles

Triangles congruent by SSS proofs

Triangles congruent by SAS and HL proofs

Triangles congruent by ASA and AAS proofs

Isosceles and equilateral triangles

Angles and trigonometry

Use sine ratio to calculate angels and side (Sin = \(\frac{o}{h})\)

Use cosine ratio to calculate angles and side (Cos =  \(  \frac{a}{h}\)  )

Combination of SohCahToa questions

Word problems relating ladder in trigonometry

Word problems relating guy wire in trigonometry

Use tangent ratio to calculate angles and side (Tan =  \(  \frac{o}{a}\)  )

Transformations

Line symmetry

Rotation symmetry and transformations

Similarity

Scale diagrams

Similar triangles

2D figures and 3D figures

Surface area of three-dimensional (3D) objects

Nets of three-dimensional objects

Tessellations using rotations

Probability

Fundamental counting principle

Fundamental Counting Principle Involving Restrictions

Permutations

Solve problems involving both permutations and combinations

Problems involving both permutations and combinations

Probability of independent events 

Probability with Venn diagrams

Vectors

Word problems on vectors

Introduction to vectors

Magnitude of a vector

Direction angle of a vector

Equivalent vectors

Adding and subtracting vectors in component form

Set Theory

Set notation

Understanding How to Use Set Notation

Set builder notation

Intersection and union of 2 sets

Intersection and Union of 2 Sets

Intersection and union of 3 sets

Intersection and Union of 3 Sets

Interval notations

Sine Rule and Cosine Rule

Law of sines

Ambiguous case: SSA triangles

Law of cosines

Applications of the sine law and cosine law

Bearings

Introduction to bearings

Bearings and direction word problems

Bearings and Direction Word Problems

Angle of elevation and depression

Logic

Inductive reasoning

Statements

Negations

Conjunctions and disjunctions

Truth tables

Conditionals

Inverses, converses, and contrapositives

Biconditionals

Deductive reasoning

Algebraic proofs

Reasoning

Consecutive integers

Number sequences

Finding the missing digit

Angles, lines and transversals

Parallel line equations

Parallel line equation

Parallel and perpendicular lines

Combination of both parallel and perpendicular line equations

Pairs of lines and angles

Parallel lines and transversals

Parallel line proofs

Perpendicular line proofs

<p>Get better math marks with our complete Geometry help, whether it&apos;s <a href="http://www.corestandards.org/Math/Content/HSG/introduction/">Geometry common core standards</a>, Geometry <a href="http://www.fldoe.org/accountability/assessments/k-12-student-assessment/end-of-course-eoc-assessments/index.stml">EOC</a>, or <a href="https://www.engageny.org/resource/geometry-module-1">EngageNY Geometry</a> (Regents Examination). We've got you covered!</p>
<p>Keeping with your class or textbook, our thorough help for Geometry includes topics such as Similar triangles, Geometry proofs, right triangle trigonometry, 3D shapes, Triangle geometry, Midpoint, Surface area of circles, Translations, and more. Learn the concepts with our video tutorials that show you step-by-step solutions to even the hardest geometry problems. Then, strengthen your understanding with tons of geometry practice.</p> 
<p>All our lessons are taught by experienced Geometry teachers. Let&apos;s finish your homework in no time, and ACE that final.</p>


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math

<h2>What is Geometry?</h2>
<p>Understanding geometry can be difficult. With so many sub-topics under the umbrella of Geometry it can be difficult to grasp what exactly is Geometry, <a href="https://www.studypug.com/blog/">why and when could ever be useful?</a> Generally speaking, Geometry is study of shapes &ndash; how they can be transformed, scaled, measuring/calculating the angles found within them, as well as other characteristics such as area, volume, and so forth. The study of Geometry also involves the study of lines, the building blocks of shapes, and their properties. Here at StudyPug, Geometry will also introduce you to the realm of statistics where we look at probability, and how this is to shapes (i.e. Venn diagrams).</p>

<h2>Who is known as the father of geometry?</h2>
<p>The honour and prestige of this title was bestowed to the one and only Euclid of Alexandria. He was an incredible Mathematician who studied at Plato&apos;s Academy in Greece and the first to study Geometry.  He went on to publish a variety of textbooks used by mathematicians to this day. He is most known for his work on a publication titled "The Elements" that feature ground-breaking content on algebra, number theories, and of course, geometry. It is said that this masterpiece was what earned him the title of "Father of Geometry" (<a href="https://en.wikipedia.org/wiki/Euclid">https://en.wikipedia.org/wiki/Euclid</a>).</p>

<h2>Is geometry hard?</h2>
<p>Studying Geometry can have its challenges and learning curves, but that&apos;s why we&apos;re here! Here at StudyPug, we offer help for all - those who fall under the category of Geometry for dummies or those who are looking for more practice and/or advanced questions. At StudyPug we strive to make learning geometry easy, accessible, and engaging. With over 1000&apos;s of lessons, Step-by-step examples, and 24/7 help, we are prepared to help students with endless opportunities for easy practice, be it for Geometry exam preparation or to simply to reinforce learning. Here at StudyPug we cover a wide variety of questions you may encounter and we also use a variety of Geometry textbooks (e.g. Prentice Hall Geometry; Glencoe Geometry; McDougal Littell Geometry; Big Ideas Math Geometry) to cater comprehensively to the content you may be learning. Rest assured, we&apos;re well equipped to help with the fundamental concepts and basic problems that will allow you to progress confidently to more complex, challenging questions at your own pace.</p>

<h2>What is Euclidean geometry?</h2>
<p>Euclidean geometry is the branch of geometry that is focused on the study of 2-dimension solid geometry and planes. Some examples include basic facts such as the interior angles of triangles should equal to 180 degrees and that the shortest distance between point A and B is a straight line. Euclidean geometry was built on the basis of Euclid mathematical theories and postulates, most notably composed and synthesized in his publication "The Elements". In a broader sense, Euclidean geometry is also referred to the importance of utilizing cognitive abilities in Math. This encompasses the learner to defer techniques of memorization and formulas, and to rely on a strong, holistic understanding of the subject that allows application of skills and techniques to solve Math problems. While you may have begun studying Euclidean geometry early on in school, you will soon begin exploring concepts of non-euclidean geometry (e.g. properties of 3D shapes, such as a sphere).</p>

<h2>How to pass geometry?</h2>
<p>Whether you&apos;re preparing for a Geometry test, exam, or simply require Geometry help, StudyPug is here to help you succeed and pass your assessments through our Geometry tutoring and Geometry lessons.  Our Geometry tutors will cover content, study tips, strategies, and practice questions that will provide you with as much help with Geometry as you need. We recommend starting with the difficult sub-topics/chapters you are struggling with, follow our Geometry tutorials as we take you step-by-step through the process of understanding a concept or problem you may have. Some concepts or problems are trickier than others so you may need to revisit theses areas more than once, which shouldn&apos;t be a problem given your unlimited, 24/7 access. Once you are feeling confident, we encourage you to practice, practice, practice! This enables you to be prepared for different styles of questions as well as different levels of difficulty. These questions will supplement your learning and truly test your understanding.</p>

<h2>What grade is geometry?</h2>
<p>Students begin learning Geometry in 10th Grade. Students should have a strong understanding of <a href="[[region_textbook.html|{"textbook_key":"algebra-1"}]]">Algebra 1</a> (completed in 9th Grade) before beginning to tackle topics covered in Geometry. For a refresher, please see our <a href="[[region_textbook.html|{"textbook_key":"algebra-1"}]]">Algebra 1 page</a>.  Following the completion of Geometry, students may proceed to beginning <a href="[[region_textbook.html|{"textbook_key":"algebra-2"}]]">Algebra 2</a>. Ahead of the game? Wonder what comes after Geometry? Visit our <a href="[[region_textbook.html|{"textbook_key":"algebra-2"}]]">Algebra 2 page</a> for more learning.</p>

https://dmn92m25mtw4z.cloudfront.net/img/textbook/textbook-geometry.png

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.   

Rotational symmetry and transformations

Learning rotational symmetry and transformations

Understanding tessellations

Tessellations using translations and reflections