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The program is really easy-to-use since most of the utilities within the suite can be controlled and accessed through the Task Launcher homepage. Microsoft Works refers to Productivity, Office Software, and supports. It should be mentioned that the setup files wkscal. The download link of the program is safe to use. Productivity Office Software Microsoft Works. Microsoft Works 9. By Microsoft Corporation. Free download. June 8th, Download Microsoft Works 9. Comments on Microsoft Works Thank you for rating the program!

NET Framework is Microsoft's comprehensive and consistent … more info More Microsoft Update Health Tools 4. Check your computer to make sure it is working properly and is compatible with the upcoming Microsoft Windows More Microsoft Silverlight 5.

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Descriptions containing microsoft works version 9. Audio chipsets from Realtek are used in motherboards from many different manufacturers. If you have such a motherboard, you can use the drivers provided by Realtek. More Google Chrome Chrome is a web browser developed by Google. It is characterized by its speed and many innovative features. More Apple Software Update 2. Apple Software Update is a software tool by Apple that installs the latest version of Apple software.

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In mathematics , the Pythagorean theorem , or Pythagoras' theorem , is a fundamental relation in Euclidean geometry among the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse the side opposite the right angle is equal to the sum of the areas of the squares on the other two sides. This theorem can be written as an equation relating the lengths of the legs a , b and the hypotenuse c , often called the Pythagorean equation : [1].

The theorem is named for the Greek philosopher Pythagoras , born around BC. Although it was attributed to him in classical antiquity , there is evidence that aspects of the theorem were known in earlier cultures; modern scholarship has also questioned whether Pythagoras himself was aware of it. The theorem has been proven numerous times by many different methods — possibly the most for any mathematical theorem.

The proofs are diverse, including both geometric proofs and algebraic proofs, with some dating back thousands of years. The theorem can be generalized in various ways: to higher-dimensional spaces , to spaces that are not Euclidean , to objects that are not right triangles, and to objects that are not triangles at all but n -dimensional solids.

The Pythagorean theorem has attracted interest outside mathematics as a symbol of mathematical abstruseness , mystique , or intellectual power; popular references in literature, plays, musicals, songs, stamps, and cartoons abound. If c denotes the length of the hypotenuse and a and b denote the two lengths of the legs of a right triangle, then the Pythagorean theorem can be expressed as the Pythagorean equation:. If only the lengths of the legs of the right triangle are known but not the hypotenuse, then the length of the hypotenuse can be calculated with the equation.

If the length of the hypotenuse and of one leg is known, then the length of the other leg can be calculated as. A generalization of this theorem is the law of cosines , which allows the computation of the length of any side of any triangle, given the lengths of the other two sides and the angle between them.

If the angle between the other sides is a right angle, the law of cosines reduces to the Pythagorean equation. In the first square, the triangles are placed such that the corners of the square correspond to the corners of the right angle in the triangles, forming a square in the center whose sides are length c. In the second square, the 4 triangles are formed into two rectangles with sides lengths a and b and placed such that a square of side length a is formed in one corner and a square of b is formed in the other.

In another proof rectangles in the second box can also be placed such that both have one corner that correspond to consecutive corners of the square. Heath himself favors a different proposal for a Pythagorean proof, but acknowledges from the outset of his discussion "that the Greek literature which we possess belonging to the first five centuries after Pythagoras contains no statement specifying this or any other particular great geometric discovery to him.

The theorem can be proved algebraically using four copies of the same triangle arranged symmetrically around a square with side c , as shown in the lower part of the diagram. The four triangles and the square side c must have the same area as the larger square,.

A similar proof uses four copies of a right triangle with sides a , b and c , arranged inside a square with side c as in the top half of the diagram. The area of the large square is therefore.

This theorem may have more known proofs than any other the law of quadratic reciprocity being another contender for that distinction ; the book The Pythagorean Proposition contains proofs.

This proof is based on the proportionality of the sides of three similar triangles, that is, upon the fact that the ratio of any two corresponding sides of similar triangles is the same regardless of the size of the triangles. Let ABC represent a right triangle, with the right angle located at C , as shown on the figure. Draw the altitude from point C , and call H its intersection with the side AB.

Point H divides the length of the hypotenuse c into parts d and e. The proof of similarity of the triangles requires the triangle postulate : The sum of the angles in a triangle is two right angles, and is equivalent to the parallel postulate. Similarity of the triangles leads to the equality of ratios of corresponding sides:. The role of this proof in history is the subject of much speculation. The underlying question is why Euclid did not use this proof, but invented another.

One conjecture is that the proof by similar triangles involved a theory of proportions, a topic not discussed until later in the Elements , and that the theory of proportions needed further development at that time. In outline, here is how the proof in Euclid 's Elements proceeds. The large square is divided into a left and right rectangle.

A triangle is constructed that has half the area of the left rectangle. Then another triangle is constructed that has half the area of the square on the left-most side.

These two triangles are shown to be congruent , proving this square has the same area as the left rectangle. This argument is followed by a similar version for the right rectangle and the remaining square.

Putting the two rectangles together to reform the square on the hypotenuse, its area is the same as the sum of the area of the other two squares. The details follow. Let A , B , C be the vertices of a right triangle, with a right angle at A. Drop a perpendicular from A to the side opposite the hypotenuse in the square on the hypotenuse. That line divides the square on the hypotenuse into two rectangles, each having the same area as one of the two squares on the legs.

For the formal proof, we require four elementary lemmata :. Next, each top square is related to a triangle congruent with another triangle related in turn to one of two rectangles making up the lower square. This proof, which appears in Euclid's Elements as that of Proposition 47 in Book 1, demonstrates that the area of the square on the hypotenuse is the sum of the areas of the other two squares. Another by rearrangement is given by the middle animation.

A large square is formed with area c 2 , from four identical right triangles with sides a , b and c , fitted around a small central square.

Then two rectangles are formed with sides a and b by moving the triangles. Combining the smaller square with these rectangles produces two squares of areas a 2 and b 2 , which must have the same area as the initial large square. The third, rightmost image also gives a proof. The upper two squares are divided as shown by the blue and green shading, into pieces that when rearranged can be made to fit in the lower square on the hypotenuse — or conversely the large square can be divided as shown into pieces that fill the other two.

This way of cutting one figure into pieces and rearranging them to get another figure is called dissection. This shows the area of the large square equals that of the two smaller ones. Albert Einstein gave a proof by dissection in which the pieces do not need to be moved.

In Einstein's proof, the shape that includes the hypotenuse is the right triangle itself. The dissection consists of dropping a perpendicular from the vertex of the right angle of the triangle to the hypotenuse, thus splitting the whole triangle into two parts. Those two parts have the same shape as the original right triangle, and have the legs of the original triangle as their hypotenuses, and the sum of their areas is that of the original triangle.

Because the ratio of the area of a right triangle to the square of its hypotenuse is the same for similar triangles, the relationship between the areas of the three triangles holds for the squares of the sides of the large triangle as well. As shown in the accompanying animation, area-preserving shear mappings and translations can transform the squares on the sides adjacent to the right-angle onto the square on the hypotenuse, together covering it exactly.

The translations also leave the area unchanged, as they do not alter the shapes at all. Each square is first sheared into a parallelogram, and then into a rectangle which can be translated onto one section of the square on the hypotenuse. A related proof was published by future U. President James A. Garfield then a U. Representative see diagram. The area of the trapezoid can be calculated to be half the area of the square, that is.

One can arrive at the Pythagorean theorem by studying how changes in a side produce a change in the hypotenuse and employing calculus.

The triangle ABC is a right triangle, as shown in the upper part of the diagram, with BC the hypotenuse. At the same time the triangle lengths are measured as shown, with the hypotenuse of length y , the side AC of length x and the side AB of length a , as seen in the lower diagram part.

If x is increased by a small amount dx by extending the side AC slightly to D , then y also increases by dy. Therefore, the ratios of their sides must be the same, that is:. This is more of an intuitive proof than a formal one: it can be made more rigorous if proper limits are used in place of dx and dy. The converse of the theorem is also true: [25]. It can be proven using the law of cosines or as follows:. Construct a second triangle with sides of length a and b containing a right angle.

Since both triangles' sides are the same lengths a , b and c , the triangles are congruent and must have the same angles. Therefore, the angle between the side of lengths a and b in the original triangle is a right angle.

The above proof of the converse makes use of the Pythagorean theorem itself. The converse can also be proven without assuming the Pythagorean theorem. A corollary of the Pythagorean theorem's converse is a simple means of determining whether a triangle is right, obtuse, or acute, as follows. The following statements apply: [29]. Edsger W. Dijkstra has stated this proposition about acute, right, and obtuse triangles in this language:. In other words, a Pythagorean triple represents the lengths of the sides of a right triangle where all three sides have integer lengths.

Some well-known examples are 3, 4, 5 and 5, 12, A primitive Pythagorean triple is one in which a , b and c are coprime the greatest common divisor of a , b and c is 1. The Pythagorean theorem has,. The reciprocal Pythagorean theorem is a special case of the optic equation. One of the consequences of the Pythagorean theorem is that line segments whose lengths are incommensurable so the ratio of which is not a rational number can be constructed using a straightedge and compass.

Pythagoras' theorem enables construction of incommensurable lengths because the hypotenuse of a triangle is related to the sides by the square root operation. The figure on the right shows how to construct line segments whose lengths are in the ratio of the square root of any positive integer. In each right triangle, Pythagoras' theorem establishes the length of the hypotenuse in terms of this unit. For more detail, see Quadratic irrational.

Incommensurable lengths conflicted with the Pythagorean school's concept of numbers as only whole numbers. The Pythagorean school dealt with proportions by comparison of integer multiples of a common subunit. So the three quantities, r , x and y are related by the Pythagorean equation,. Note that r is defined to be a positive number or zero but x and y can be negative as well as positive. Geometrically r is the distance of the z from zero or the origin O in the complex plane.

   

 

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Apple Inc. Apple went public in , to instant financial success. The company developed computers featuring innovative graphical user interfaces , including the original Macintosh , announced in a critically acclaimed advertisement, " ", directed by Ridley Scott.

By , the high cost of its products and power struggles between executives caused problems. Wozniak stepped back from Apple amicably and pursued other ventures, while Jobs resigned bitterly and founded NeXT , taking some Apple employees with him.

As the market for personal computers expanded and evolved throughout the s, Apple lost considerable market share to the lower-priced duopoly of the Microsoft Windows operating system on Intel -powered PC clones also known as " Wintel ". In , weeks away from bankruptcy, the company bought NeXT to resolve Apple's unsuccessful operating system strategy and entice Jobs back to the company.

Over the next decade, Jobs guided Apple back to profitability through a number of tactics including introducing the iMac , iPod , iPhone and iPad to critical acclaim, launching " Think different " and other memorable advertising campaigns, opening the Apple Store retail chain, and acquiring numerous companies to broaden the company's product portfolio.

Apple became the first publicly traded U. The company receives criticism regarding the labor practices of its contractors, its environmental practices, and its business ethics, including anti-competitive practices and materials sourcing.

Nevertheless, the company has a large following and enjoys a high level of brand loyalty. It is ranked as one of the world's most valuable brands. Apple Computer, Inc. The Apple II was chosen to be the desktop platform for the first " killer application " of the business world: VisiCalc , a spreadsheet program released in A critical moment in the company's history came in December when Jobs and several Apple employees, including human—computer interface expert Jef Raskin , visited Xerox PARC in to see a demonstration of the Xerox Alto , a computer using a graphical user interface.

Xerox granted Apple engineers three days of access to the PARC facilities in return for the option to buy , [39] shares 5. The Lisa division would be plagued by infighting, and in Jobs was pushed off the project. The Lisa launched in and became the first personal computer sold to the public with a GUI, but was a commercial failure due to its high price and limited software titles. Jobs, angered by being pushed off the Lisa team, took over the company's Macintosh division. Wozniak and Raskin had envisioned the Macintosh as low-cost-computer with a text-based interface like the Apple II, but a plane crash in forced Wozniak to step back from the project.

Jobs quickly redefined the Macintosh as a graphical system that would be cheaper than the Lisa, undercutting his former division. In , Apple launched the Macintosh, the first personal computer to be sold without a programming language.

The advertisement created great interest in the original Macintosh , and sales were initially good, but began to taper off dramatically after the first three months as reviews started to come in.

The board of directors instructed Sculley to contain Jobs and his ability to launch expensive forays into untested products. Rather than submit to Sculley's direction, Jobs attempted to oust him from his leadership role at Apple.

PageMaker , an early desktop publishing application taking advantage of the PostScript language, was also released by Aldus Corporation in July This dominant position in the desktop publishing market [67] allowed the company to focus on higher price points, the so-called "high-right policy" named for the position on a chart of price vs.

Newer models selling at higher price points offered higher profit margin , and appeared to have no effect on total sales as power users snapped up every increase in speed. This policy began to backfire in the last years of the decade as desktop publishing programs appeared on PC clones that offered some or much of the same functionality of the Macintosh, but at far lower price points. The company lost its dominant position in the desktop publishing market and estranged many of its original consumer customer base who could no longer afford their high-priced products.

The company pivoted strategy and in October introduced three lower-cost models, the Macintosh Classic , the Macintosh LC , and the Macintosh IIsi , all of which saw significant sales due to pent-up demand.

The same year, Apple introduced System 7 , a major upgrade to the Macintosh operating system, adding color to the interface and introducing new networking capabilities. The success of the lower-cost Macs and PowerBook brought increasing revenue. The magazine MacAddict named the period between and as the "first golden age" of the Macintosh. The success of Apple's lower-cost consumer models, especially the LC, also led to the cannibalization of their higher-priced machines.

To address this, management introduced several new brands, selling largely identical machines at different price points, aimed at different markets: the high-end Quadra models, the mid-range Centris line, and the consumer-marketed Performa series.

This led to significant market confusion, as customers did not understand the difference between models.

The early s also saw the discontinuation of the Apple II series , which was expensive to produce, and the company felt was still taking sales away from lower-cost Macintosh models. After the launch of the LC, Apple began encouraging developers to create applications for Macintosh rather than Apple II, and authorized salespersons to direct consumers towards Macintosh and away from Apple II. Throughout this period, Microsoft continued to gain market share with its Windows graphical user interface that it sold to manufacturers of generally less expensive PC clones.

While the Macintosh was more expensive, it offered a more tightly integrated user experience, but the company struggled to make the case to consumers. Apple also experimented with a number of other unsuccessful consumer targeted products during the s, including digital cameras , portable CD audio players , speakers , video game consoles , the eWorld online service, and TV appliances. Most notably, enormous resources were invested in the problem-plagued Newton tablet division, based on John Sculley's unrealistic market forecasts.

Throughout this period, Microsoft continued to gain market share with Windows by focusing on delivering software to inexpensive personal computers, while Apple was delivering a richly engineered but expensive experience. Microsoft Corp. The major product flops and the rapid loss of market share to Windows sullied Apple's reputation, and in Sculley was replaced as CEO by Michael Spindler. In the wake of the alliance, Apple opened up to the idea of allowing Motorola and other companies to build Macintosh clones.

Over the next two years, 75 distinct Macintosh clone models were introduced. However, by Apple executives were worried that the clones were cannibalizing sales of their own high-end computers, where profit margins were highest.

Hired for his reputation as a corporate rehabilitator, Amelio made deep changes, including extensive layoffs and cost-cutting.

This period was also marked by numerous failed attempts to modernize the Macintosh operating system MacOS. The original Macintosh operating system System 1 was not built for multitasking running several applications at once. The company attempted to correct this with by introducing cooperative multitasking in System 5, but the company still felt it needed a more modern approach.

On July 9, , Jobs staged a boardroom coup that resulted in Amelio's resignation after overseeing a three-year record-low stock price and crippling financial losses. The board named Jobs as interim CEO and he immediately began a review of the company's products.

On May 6, , Apple introduced a new all-in-one computer reminiscent of the original Macintosh: the iMac. The device also had a striking eardrop shape and translucent materials, designed by Jonathan Ive , who although hired by Amelio, would go on to work collaboratively with Jobs for the next decade to chart a new course the design of Apple's products. A little more than a year later on July 21, , Apple introduced the iBook , a laptop for consumers.

It was the culmination of a strategy established by Jobs to produce only four products: refined versions of the Power Macintosh G3 desktop and PowerBook G3 laptop for professionals, along with the iMac desktop and iBook laptop for consumers. Jobs felt the small product line allowed for a greater focus on quality and innovation.

At around the same time, Apple also completed numerous acquisitions to create a portfolio of digital media production software for both professionals and consumers. Apple acquired of Macromedia 's Key Grip digital video editing software project which was renamed Final Cut Pro when it was launched on the retail market in April Apple renamed the program iTunes , while simplifying the user interface and adding the ability to burn CDs. The first announcement came on March 24, , that Apple was nearly ready to release a new modern operating system, Mac OS X.

The announcement came after numerous failed attempts in the early s, and several years of development. In May , the company opened its first two Apple Store retail locations in Virginia and California, [] [] offering an improved presentation of the company's products. On October 23, , Apple debuted the iPod portable digital audio player.

The product, which was first sold on November 10, , was phenomenally successful with over million units sold within six years. In , Apple's iTunes Store was introduced. The iTunes Store quickly became the market leader in online music services, with over five billion downloads by June 19, In , Apple purchased Nothing Real for their advanced digital compositing application Shake , [] as well as Emagic for the music productivity application Logic.

The purchase of Emagic made Apple the first computer manufacturer to own a music software company. The acquisition was followed by the development of Apple's consumer-level GarageBand application.

By August 7, , Apple made the transition to Intel chips for the entire Mac product line—over one year sooner than announced. Apple's success during this period was evident in its stock price. In an article posted on Apple's website on February 6, , Jobs wrote that Apple would be willing to sell music on the iTunes Store without digital rights management DRM , thereby allowing tracks to be played on third-party players, if record labels would agree to drop the technology.

On January 14, , Jobs announced in an internal memo that he would be taking a six-month medical leave of absence from Apple until the end of June and would spend the time focusing on his health. In the email, Jobs stated that "the curiosity over my personal health continues to be a distraction not only for me and my family, but everyone else at Apple as well", and explained that the break would allow the company "to focus on delivering extraordinary products".

After years of speculation and multiple rumored "leaks", Apple unveiled a large screen, tablet-like media device known as the iPad on January 27, The iPad ran the same touch-based operating system as the iPhone, and all iPhone apps were compatible with the iPad. This gave the iPad a large app catalog on launch, though having very little development time before the release.

Later that year on April 3, , the iPad was launched in the US. It sold more than , units on its first day, and , by the end of the first week. In June , Apple released the iPhone 4 , [] [] which introduced video calling using FaceTime , multitasking , and a new uninsulated stainless steel design that acted as the phone's antenna.

Later that year, Apple again refreshed its iPod line of MP3 players by introducing a multi-touch iPod Nano , an iPod Touch with FaceTime , and an iPod Shuffle that brought back the clickwheel buttons of earlier generations. On January 17, , Jobs announced in an internal Apple memo that he would take another medical leave of absence for an indefinite period to allow him to focus on his health.

Chief Operating Officer Tim Cook assumed Jobs's day-to-day operations at Apple, although Jobs would still remain "involved in major strategic decisions". Apple did not have a chairman at the time [] and instead had two co-lead directors, Andrea Jung and Arthur D. Levinson , [] who continued with those titles until Levinson replaced Jobs as chairman of the board in November after Jobs' death.

On October 5, , Steve Jobs died, marking the end of an era for Apple. From to , Apple released the iPhone 4S [] [] and iPhone 5 , [] [] which featured improved cameras, an intelligent software assistant named Siri , and cloud-synced data with iCloud; the third and fourth generation iPads, which featured Retina displays ; [] [] [] and the iPad Mini , which featured a 7.

This beat the non-inflation-adjusted record for market capitalization previously set by Microsoft in In May , the company confirmed its intent to acquire Dr. Iovine believed that Beats had always "belonged" with Apple, as the company modeled itself after Apple's "unmatched ability to marry culture and technology. During a press event on September 9, , Apple introduced a smartwatch, the Apple Watch. In January , it was announced that one billion Apple devices were in active use worldwide.

On June 6, , Fortune released Fortune , their list of companies ranked on revenue generation. In the trailing fiscal year , Apple appeared on the list as the top tech company. In November , Apple announced it was branching out into original scripted programming: a drama series starring Jennifer Aniston and Reese Witherspoon , and a reboot of the anthology series Amazing Stories with Steven Spielberg.