Improved Software Updater now supports an additional 50 applications (soon to increase to 200), including KMplayer, WinZip, Box Drive, Foxit Reader, Adobe Reader, and Nitro Pro. Performance Optimizer intelligently puts these apps ‘to sleep’ in order to redistribute resources to the task at hand, and then wakes them up instantly as soon as they’re needed.ĥ0 new programs added to Software UpdaterĬCleaner Professional’s updated Software Updater ensures users get the latest version of the programs they use by applying new updates and patches which can help fix bugs and security vulnerabilities provided by the app or device provider. PCs use apps to run start-up items, background processes, and services invisibly in the background while not in use, which slows down the performance and running of machines. It makes old PCs feel new again and new PCs stay fast, no matter how many programs are installed. It is highly likely that this software program is malicious or has security issues or other reasons.Performance Optimizer: A patent for day 1 performanceĬCleaner 6.0 Professional comes with Performance Optimizer, which includes patented technology that stops programs the user isn’t working with, draining a PC’s resources. This software is no longer available for download. These could be false positives, and our users are advised to be careful when installing and using this software. These may include a toolbar, changing your homepage, default search engine, or installing other party programs. This program is advertising supported and may offer to install third-party programs that are not required. And there is no third-party softwarebundled. We scanned files and URLs associated with this software program in more than 60 of the world's leading antivirus services no possible threats were found. It is highly likely that this software program is clean. Based on this checking, we set the following security levels for any downloadable files. To avoid any potential harm to your device and ensure the security of your data and privacy, our team checks every time a new installation file is uploaded to our servers or linked to a remote server and periodically reviews the file to confirm or update its status.
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If you take a bundle with both Virgin Media and O2, you’ll have separate contracts and bills – one for Virgin Media, and another for O2. TV 360 and home phone not currently available in 2022/23 network expansion areas.Īll of the products on this page are subject to survey, network capacity and a credit check.Įach service has a minimum contract length that’s shown in your basket. Virgin Media services are only available in eligible Virgin Media network areas. If you can set up using a Self-Install pack, but choose to use an engineer for the installation, you’ll still be charged £30 for this service. No setup fee is available on all bundles, broadband & phone, and broadband only packages. This promotion means you won’t pay the usual setup fee of £35. Doesn’t apply to any add-ons you may have selected, out-of-bundle charges or admin costs, which may increase separately. *Monthly price of Virgin Media’s main services and O2 Airtime Plan will increase each April from April 2024 by the Retail Price Index rate of inflation announced in February each year plus 3.9%. The internet may be faster in the mornings or on weekdays when fewer people are online, compared to busy peak-times when there’s high local demand for the internet (around 8-10pm or during big events e.g. If you’re working from home using a VPN, this extra level of protection could be affecting your broadband speed, especially during peak office hours. Try disconnecting and reconnecting to the VPN (Virtual Personal Network). Unlike an Ethernet connection, WiFi signal can be disturbed by obstacles such as walls, furniture and doors. Using an Ethernet cable to connect devices, like your online gaming console or laptop, to the Hub is a reliable way to get fast speeds. If a device is still struggling with WiFi speed, check the latest software is installed and update your internet browser – quick note: newer devices have better reception than older ones. Run our clever tests to find and fix issues with your home’s connection and check for issues in your area.ĭisconnect devices around the home that aren’t currently being used to try to speed up the internet connection and boost WiFi signal. Turning the Hub off and on again to reboot it, every two months, helps correct slow speeds caused by buffers. Positioned at least half a metre away from objects such as TVs, wireless speakers, cordless phones and radiators (keep the Hub cool). But if you’re getting broadband speed test results that are close to our estimated speeds for your package and you’re finding that the connection is too slow, it may be time to upgrade your broadband to a faster speed. You can also follow our guide on how to improve latency. These tips and tricks can help improve connection and combat slower speeds. Test your broadband speed at different times of day for an average reading.Check your device is not too old to be compatible with high broadband speeds.Switch off antivirus software and firewalls, if possible.Disconnect your other devices or consoles from the WiFi as they can interfere with the signal.Close all other browser tabs, windows and apps.Ways to make your internet speed check more accurate But using an Ethernet cable to connect your device to the Hub will give you the most accurate broadband speed test results and a faster connection than wireless broadband. Obviously, you can run a WiFi test to check what speeds you’re getting around your home. And if your results vary, don’t worry that’s normal, due to factors such as the time of day and number of devices sharing your broadband when you repeat the internet speed test. If anything, it’s more likely to underestimate speeds – it’s a strict, Ofcom-approved test (other speed checkers may not be). Our broadband speed test is generally pretty accurate and the results are a good reflection of the speeds you were getting during the test. In contrast, the G♭ trigger adds a standard 5th position length. Pressing the F trigger equates to a standard 6th position length. Each unit would likely be close to 1 cm in length. Notice that by the time we get down to 12 th position, the distance between positions has almost doubled!įor this table, I have assumed the distance between 1 st and 2 nd position to be 10 units for simplicity. This chart shows the length of each successive slide position increasing by 6%. If we think of the slide positions continuing on from 7 th position by utilizing triggers, we can make the instrument long enough to play a low B♮ in what is essentially 12 th position. As your trombone extends, positions drift farther apart by 6% for each position. The gap between the 2nd and 3rd positions is 6% wider than that between the 1st and 2nd positions, and this pattern continues. Let’s get started… The Physics of Slide Positions for Double Trigger TrombonesĪ fundamental concept is that to produce a note one semitone lower, the length of the instrument must increase by 6%. In this article, we will go more in depth. Our article THE TROMBONE TRIGGER MAKES IT LONGER discusses the mechanics behind slide length adjustments using triggers. Cracking the Code: Two Solutions for PrecisionĮmbrace Technology with Electronic Tuners: Using an electronic tuner can help guide you to the correct slide positions with any trigger combination.ĭelving into Mechanics: For those who relish understanding the “why” behind things, this method is for you. So, 2nd position, for example, will vary depending on whether one, two, or no triggers are being pressed. But activate both triggers, and the positions stretch further apart than they do with only one trigger engaged. Pressing a single trigger elongates the slide positions compared to playing without any triggers. Greetings to all trombonists exploring the realm of double trigger instruments! If you’ve been on the hunt for a slide position chart tailored to these bass trombones, in this guide, we’ll delve into why a standard chart might not be the ultimate solution you’re hoping for, and we’ll uncover methods to discover slide positions for double trigger trombones. School districts are required to establish and implement procedural safeguards with respect to identification, evaluation, and placement procedures. If the school determines the behavior is not a manifestation of the student’s disability, the student will be subject to the same disciplinary actions (including suspensions or expulsion) that apply to students without disabilities. If the student’s behavior is a manifestation of their disability, the student should be allowed to return to their school placement. A manifestation determination meeting brings together the 504 team to assess whether the student’s behavior that led to discipline is linked to their disability or a failure to implement their 504 plan. suspensions, etc.) for more than 10 days, a manifestation determination meeting must be held. If a section 504 student is subject to discipline (i.e. Least Restrictive Environment refers to the right of students with disabilities to be educated as much as possible with children who do not have disabilities.Ī school may discipline a student on a 504 plan for fewer than 10 days in the same way they would discipline students without disabilities. The right to a Free Appropriate Public Education (FAPE) generally means that children who have disabilities, including children with disabilities who have been suspended or expelled from school, have the right to be educated at public expense, in a manner appropriate to meet their unique needs. What rights does section 504 provide for students?įree and Appropriate Education in the Least Restrictive Environmentįederally funded schools must provide a free and appropriate public education in the least restrictive environment to all students who have a disability. A student’s 504 plan will provide accommodations that allow a student with an impaired major life activity to have the same level of access to the instruction, school activities, and school building as students without disabilities. However, a student with a 504 plan requires accommodations in order to gain equal access to instruction and/or the school facility. Unlike a special education student, a student with a 504 plan is able to make effective progress in school without the need for specialized instruction and/or related services. In order to be found eligible for special education services and an Individualized Education Program (IEP), a student must be diagnosed with a disability and must, as a result of that disability, require specialized instruction and/or related services in order to make effective progress. How is a 504 plan different from special education? Students may be eligible for a 504 plan if a temporary disability limits a major life activity. The disability must substantially limit one or more major life activities. The student must have information establishing a mental, psychological, and/or physical disability (including a disability affecting behavior) and In order to be eligible for a 504 plan, a student must attend a school that receives federal financial assistance (which includes all public schools in Massachusetts). How does a student qualify for a section 504 plan? Section 504 requires consent from the parents before conducting evaluations. In addition, the district must believe that the student needs additional supports in the regular education classroom in order access the curriculum. Major life activities include walking, seeing, hearing, speaking, and learning. The 504 plan process may be started by a parent’s written request or the school’s referral for an evaluation when the student is having academic, social, or behavioral problems that limit one or more major life activities. This includes school-sponsored non-academic and extracurricular services and activities. Students who qualify for a 504 plan must be given the opportunity to receive any help, benefit, and services and participate in activities that are available for students without disabilities. Section 504 ensures that a student with a disability has equal access to education by requiring that schools provide accommodations for the student. Section 504 is a section of a federal civil rights law, the Rehabilitation Act of 1973, that prohibits discrimination by federally-funded institutions, such as public schools, against individuals with disabilities. It said that the watch was still likely to be auctioned and the climate initiative “will properly report it, as all of Arnold’s nonprofits do.” Schwarzenegger's charity auctions raise millions of dollars every year for after-school programs for children across the U.S. It added in an emailed statement that Schwarzenegger “cooperated at every step even though it was an incompetent shakedown” and that he agreed to “prepay potential taxes on the watch (remember, it is his personal watch).” The Schwarzenegger Climate Initiative said that Schwarzenegger was detained “for traveling with a watch he owns, that he might be auctioning at his charity auction (Thursday) in Kitzbuehel,” in neighboring Austria. German daily Bild, which first reported on the incident, reported that the allegedly undeclared watch made by Swiss manufacturer Audemars Piguet was valued at about 20,000 euros ($21,739) by Schwarzenegger, for whom it was specially produced. Goods over the value of 430 euros ($467) that will stay in the European Union have to be declared and, where appropriate, duty paid on them. Schwarzenegger was able to leave after about 2½ hours, he said Thursday. The former movie star and California governor was stopped for a routine check after arriving Wednesday, customs spokesperson Thomas Meister said. BERLIN – Arnold Schwarzenegger was stopped for hours by customs at Munich Airport after entering Germany with a luxury watch that was potentially to be auctioned at a charity event. To play this interactive game, you will need to fill empty spaces of a 9×9 and 4×4 grid. They help sharpen mind and improve concentration skills. Puzzles like Sudoku help to keep our brain challenged and engaging. I hope you enjoy this collection of 100 large-size sudoku printables.Looking to improve your brain health? Studies suggest that solving puzzles are highly effective for brain functionality and improving memory. It's great for sharpening brain skills such as concentration and numerical reasoning. Sudoku is one of my favorite number puzzles. See Kevin Rush's sudoku strategy guides for a detailed explanation of this and other tricks. That way you won't waste time checking numbers you've finished searching. Once you've found all instances of a number, cross that number off your list. This is known as tagging.Īnother is to list the numbers 1 through 9 on the side of the puzzle. One common strategy is to enter guesses in small writing in the cells. You are allowed to make notes and other marks on the sudoku printable. I recommend using a pencil with an eraser, since it's quite typical when solving sudoku puzzles to make a few errors along the way. (See the sudoku puzzle image at left as an example.)Īs I mentioned above, you need to enter a single digit 1 through 9 in each of the 81 small squares in the puzzle. The grid is further subdivided into 3-by-3 boxes. These small squares are also known as cells. Sudoku puzzles consist of a large square divided into a 9-by-9 grid. Every sudoku puzzle has only a single solution. The puzzle is solved once you have filled in all the small squares following this rule. Enter a number 1 through 9 in each cell of the puzzle such that no number is repeated in any row, column, or 3-by-3 square. You'll learn advanced techniques such as Ghosting, RC Counting, Tagging, and Line Checking. Note: To learn powerful strategies for solving sudoku, see Kevin Rush's free guide here. MORE PRINTABLES: Crossword Puzzles, Word Searches, Bible Word Searches, Special Needs Word Searches, Mormon Word Searches, Balance Quest Math Puzzles Review of Sudoku Rulesįor those new to sudoku, I'll quickly go over the rules. Your readership will love it! (In your publication, simply include the following copyright statement next to the puzzle: "©. I hope you enjoy solving them.Īre you the publisher of a newsletter, bulletin, newspaper, or magazine? Include a sudoku puzzle or word search puzzle in each issue. The new larger sudoku printables here correct all those issues. Folks with eyesight problems may have trouble reading the numbers on those. This doesn't leave much room for annotating each puzzle with tag numbers as you try to solve it.Īlso, the four-to-a-page can feel a little cramped. On the down side, I had to shrink those puzzles so all four would fit on a single page. The main benefit is you get four sudoku puzzles one sheet, which saves paper. I thought four-sudoku-to-a-page was a good idea at the time.Ĭramming four on a page does have its advantages. More About the Free Printable Sudoku PuzzlesĪs mentioned, the free sudoku printables linked above are one to a page, unlike the other collection. The solutions themselves are huge, too, for easy viewing. The solutions (answers) to the large-print sudoku puzzles are in the back of each book. The big cells in the puzzle grid make it easier to write the number in the box. A very large-print puzzle is ideal for seniors or anyone with vision issues.Įven someone with good vision can benefit, if they have tremors from Parkinson's disease or other conditions. That makes the puzzles much easier to see and work with compared to most puzzle books. The big puzzles in these books are one puzzle to a page. The other four books are 100 Easy, 100 Medium, 100 Hard, and 100 Very Hard large-print sudoku. Hard #1 includes 25 Easy, Medium, Hard, and Very Hard sudoku puzzles. The puzzle book shown at left, Easy to V. The books are available for purchase on. The sudoku grids in these books are as large as I could possibly fit on the 8.5"x11" page. I've created a series of five large-print sudoku books that I call "Huge Print Sudoku". Tired of printing out large sudoku puzzle pages one by one? Buy my awesome sudoku puzzles books instead! Postal Service switching out ‘arrow keys’ for electronic locks They’re ready to minimize any disruption and keep your mail as secure as a treasure chest! U.S. Losing an Arrow key is serious business, but USPS has got it covered. You might get a mail, notice, or digital message explaining the situation and what they’re doing about it. Sometimes, if there’s a significant risk, the USPS will let customers know what’s up. They jump into action, changing locks and ramping up surveillance on affected routes to keep things secure. The USPS doesn’t just sit back and twiddle their thumbs. You might be thinking, “Wait, could someone get into my mailbox?” It’s a valid concern because delays might also cause a bit of a hiccup if you’re waiting for important mail. It’s all about keeping the mail service safe and sound. Routes may have to be shuffled and schedules tweaked, potentially causing a bit of a delay. The Snags in Mail Deliveryįirst up, mail delivery and collection could hit a temporary snag. If a mail carrier loses an Arrow key, it’s not just a problem for them – it can impact postal services and you, the customer. The Ripple Effect of Losing an Arrow (Master) Key This video argues that procedures may not be followed to the letter all the time, according to a former USPS Office manager. Thanks to their procedures and swift action, they keep risks low and continue earning your trust.īut are procedures followed all the time. The USPS takes the loss of an Arrow key seriously, like losing a piece in a chess game. The key is to stick to the rules and always put safety first. In severe cases, they could lose their job. They could be counseled, retrained, or even suspended. Consequences for the Mail Carrierīut what happens to the mail carrier who lost the key? Well, it depends. If needed, they may switch out the locks for an extra layer of security. The USPS Postal Inspection Service or the Office of Inspector General may even step in. The goal? Find out how they lost the key, where it was last seen, and if anyone’s used it. With the incident logged, it’s time for some detective work. The team then works out which mailboxes the key could open and how big the security risk is. They record the incident and sound the security alarm. This quick action sets the USPS’s lost key plan into motion. Time to Alert the BossĪs soon as a mail carrier realizes their Arrow key is gone, they need to let their supervisor know. Let’s see how USPS handles these situations. Losing an Arrow key is like dropping your ice cream on a hot day – it’s a big deal. The Procedure for Mail Carriers Losing Arrow (Master) Keys They’re workhorses that ensure mail security and make life easier for mail carriers. In a nutshell, Arrow keys are more than metal trinkets. It’s not just about preventing theft – it’s also about maintaining a trusted relationship with you, the customer. They’re exclusive to USPS staff, so your mail is always in safe hands. It speeds things up, making their routes more efficient and keeping customers happy. Instead of juggling dozens of keys, they only need this one. These keys are lifesavers for mail carriers. It’s a tough job, but those Arrow keys need top-level security! Mail Carriers’ Best Friends Imagine the chaos if these keys went astray! The USPS logs key movements, stores them safely, and has a full-on plan if they lose one. After a busy day, it’s returned safely, ready for the next day’s adventures. This master key isn’t just a key it’s like a skeleton key opening multiple locks.įrom centralized mail receptacles to cluster mailboxes, this key does it all. In the U.S., postal workers have a handy tool called the Arrow key, or the Master key. Postal Service switching out ‘arrow keys’ for electronic locksīackground and Purpose of Arrow (Master) KeysĮver wondered about those special keys your mail carriers use? Let’s dive into the world of Arrow keys and understand why they’re so crucial for the mail world. The Ripple Effect of Losing an Arrow (Master) Key.The Procedure for Mail Carriers Losing Arrow (Master) Keys.Background and Purpose of Arrow (Master) Keys. The cookies is used to store the user consent for the cookies in the category "Necessary". This cookie is set by GDPR Cookie Consent plugin. The cookie is used to store the user consent for the cookies in the category "Other. The cookie is set by GDPR cookie consent to record the user consent for the cookies in the category "Functional". The cookie is used to store the user consent for the cookies in the category "Analytics". These cookies ensure basic functionalities and security features of the website, anonymously. Necessary cookies are absolutely essential for the website to function properly. When you make a purchase using links on our site, we may earn an affiliate commission. Readers like you help support iGeeksBlog. Best macOS Monterey hidden features to try on your Mac.Download new Apple Macbook Pro 2021 wallpapers.Download WWDC 2021 wallpapers for iPhone and Mac.What do you think? Share your thoughts in the comments below. I’m eagerly awaiting more macOS Monterey wallpapers, but in the meantime, I’m enjoying the look of this one. macOS Monterey light wallpaper macOS Monterey dark wallpaper This means you can use the below macOS Monterey wallpapers on as small a screen as your iPhone to as big a screen as the latest 24″ iMac with a 4.5k display. The wallpapers are available in 6K resolution (6016 X 6016) and are perfect for all the screens boasting up to 6K display. You can tap either the image or the download button below it and then set it as your background via Photos or Settings on your iPhone, iPad, or iPod touch. While you’ll have to wait for the official public release later this year to enjoy the features, you can download the macOS Monterey wallpapers right away. However, the first beta version comes with only one wallpaper available in light and dark variants. The update brings exciting new features like Shortcuts on Mac, Universal Control, Quick Note, enhanced Safari, and cross-device continuity features to Mac. Please note this is a 1.8GB zipped download with 599 images.Apple released the latest macOS Monterey at WWDC 2021. Various promotional wallpapers used by Apple over the years macOS 10.13 High Sierra (5K & 6K Resolution) OS X 10.11 El Capitan (5K & 6K Resolution) OS X 10.10 Yosemite (5K & 6K Resolution) OS X 10.9 Mavericks (5K & 6K Resolution) OS X 10.8 Mountain Lion (5K & 6K Resolution) Mac OS X 10.6 Snow Leopard (5K & 6K Resolution) Mac OS X 10.5 Leopard (5K & 6K Resolution) Mac OS X 10.4 Tiger (5K & 6K Resolution) Mac OS X 10.3 Panther (5K & 6K Resolution) Mac OS X 10.2 Jaguar (5K & 6K Resolution) Mac OS X 10.0 Cheeta (5K & 6K Resolution) Mac OS 9 (Multiple Resolutions included, certain images must be tiled to appear correctly) Mac OS 8 (Multiple Resolutions included, certain images must be tiled to appear correctly) Mac OS 7.6 (Must be tiled to appear correctly) It is HIGHLY SUGGESTED to not replace or modify the contents of this folder with the Desktop Pictures folder here and instead you should just add the folder using "Add Folder or Album" setting within the Wallpaper system settings section. **WARNING FOR MACOS 13+ USERS: Please note that on macOS 13 Ventura and macOS 14 Sonoma, Apple has changed the way wallpapers are stored in the Desktop Pictures folder, changing the files from. Installation: Copy content of "Desktop Pictures" folder to ~/System/Library/Desktop Pictures/ (**SEE NOTE BELOW FOR MACOS 13+) to have all wallpapers show up natively in the default wallpaper location, or place the "Desktop Pictures" folder anywhere on your computer and add the folder on using "Desktop & Screen Saver" prefpane. Thanks again to everyone who helped in some way with this project! This will continue to be updated as needed with new OS releases that bring new wallpapers, as well any new discoveries that might have been made. Updated Post: Thanks to the strong community we have here at MacRumors, we have been able to track down and include every Wallpaper/Desktop Picture from every version of MacOS to date. I've always thought that it would be awesome to have a complete collection of every desktop wallpaper included with every Mac OS version, ranging from the earliest version of System 7 with wallpaper support all the way through to macOS High Sierra, spanning in total 27 major OS releases and 27 years between the initial release of System 7 and High Sierra. Original Post: So this isn't exactly PowerPC related, but being that this involves PPCs and that this community usually is open to helping out with fun stuff like this, I figured I'd post this thread here. By the 890s, he was called the “ king of the English.”ĭespite the title, Alfred the Great never ruled the eastern and northern regions of modern England. Alfred the Great, who was King of Wessex, negotiated a treaty in 886 that extended his rule to areas of West Mercia and Kent. Before then, England included seven Anglo-Saxon kingdoms. This could indicate a good starting point in your family tree that you can focus on.Ī string of kings and queens have ruled England since the late 880s. Keep an eye out for any surnames you recognize or have in common. If you want to find out whether you could be related to UK royalty, taking a look at previous British monarchs is a logical place to begin. Also, Ancestry has a helpful collection of royalty, nobility, and heraldry resources that you can search through. Aside from finding a connection to royalty in your tree, you may want to spend some time checking the relationship links between you and the royal person.Īrticles like this one from Legacy Tree and verified lists of gateway ancestors, like this one from the Order of the Crown of Charlemagne, can help you do your checking. Because many people throughout history have wanted to be part of royal lines, it is common to come across "bogus" gateway ancestors, whose information or connections have been faked. It is ideal to have gateway ancestors that descend from well-documented and researched lineages, meaning that their connection to a royal family can be verified. These ancestors can be hugely important in finding out whether you are in some way connected or related to the royal family. Often you will find them in around the 17th century. The term “gateway ancestors” in genealogy refers to ancestors who directly descend from royalty or nobility, thus providing a “gateway” into the royal bloodline. The FamilySearch Family Tree might even connect you directly into one of the royal lines without much digging! It all depends on how much research others have done on shared ancestors. If you don't have a FamilySearch account, set one up for free and see if any of your relatives have a match. The shared Family Tree data on might be able to tell you more about your family lines than you already know. Start looking in your current family tree, and trace your family further back with the help of historical records. With centuries of information on British royal family trees, historical records might reveal royal names in your family tree. Here’s how to hunt for the royal lines in your family tree. Even if you aren’t technically connected to the royal family tree, you’ll find a rich supply of information on your family roots along the way. In this blog post, we’ll delve into the history of the British royal family, and explore how you can find out whether you have royal ancestors. Dive into your family tree, and you may find proof of what you’ve always known deep down-you are royalty! If you have British ancestors, there’s a chance that you could have royalty somewhere in your bloodline. It’s a question that many may wonder at some time in their life: “Do I have royal blood?” Finding out that you have royal ancestry can be an exciting discovery a person can make while delving into their family history. So the image (that is, point B) is the point (1/25, 232/25). So the intersection of the two lines is the point C(51/50, 457/50). Now we need to find the intersection of the lines y = 7x + 2 and y = (-1/7)x + 65/7 by solving this system of equations. Step 2: Extend the line segment in the same direction and by the same measure. Since the reflection line is perfectly horizontal, a line perpendicular to it would be perfectly vertical. So the equation of this line is y = (-1/7)x + 65/7. Step 1: Extend a perpendicular line segment from A to the reflection line and measure it. 11) x y Q N R E Q N R E rotation 90° clockwise about the origin 12) x y S U X T S U X T rotation 180° about the origin 13) x y V Z T V Z T rotation 180° about the origin 14) x y H Y T H Y T rotation 180° about the origin-2-Create your own worksheets like this one with Infinite Pre-Algebra. So the desired line has an equation of the form y = (-1/7)x + b. Write a rule to describe each transformation. Since the line y = 7x + 2 has slope 7, the desired line (that is, line AB) has slope -1/7 as well as passing through (2,9). And since a rotation matrix commutes with its transpose, it is a normal matrix, so can be diagonalized. We are guaranteed that the characteristic polynomial will have degree n and thus n eigenvalues. So we first find the equation of the line through (2,9) that is perpendicular to the line y = 7x + 2. In Euclidean geometry, a rotation is an example of an isometry. In other words, switch x and y and make y negative. Then, using the fact that C is the midpoint of segment AB, we can finally determine point B.Įxample: suppose we want to reflect the point A(2,9) about the line k with equation y = 7x + 2. The most common rotations are 180 or 90 turns, and occasionally, 270 turns, about the origin, and affect each point of a figure as follows: Rotations About The Origin 90 Degree Rotation When rotating a point 90 degrees counterclockwise about the origin our point A (x,y) becomes A' (-y,x). A positive degree rotation runs counter clockwise and a negative degree rotation runs clockwise. Then we can algebraically find point C, which is the intersection of these two lines. What are Rotations Rotations are a type of transformation in geometry where we take a point, line, or shape and rotate it clockwise or counterclockwise, usually by 90,180, 270, -90, -180, or -270. So we can first find the equation of the line through point A that is perpendicular to line k. Note that line AB must be perpendicular to line k, and C must be the midpoint of segment AB (from the definition of a reflection). Let A be the point to be reflected, let k be the line about which the point is reflected, let B represent the desired point (image), and let C represent the intersection of line k and line AB. |
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