Group 1
Write a C++ program or each of the following problems:
1. Write a C++ program to enter a distance in meters and print out its value in kilometers, yards, and miles. (Note: 1 m = 0.001 km = 1.094 yd = 0.0006215 mi). 2. Write a C++ program to enter length and width of a rectangle, compute and print the area and perimeter of the rectangle. Print both rounded to the nearest tenth of a foot. 3. Write a program to compute the cost for carpeting a room. Input should consist of the room length (in meters), room width (in meters), and carpet price per square meter. 4. Compute and print the number of minutes in a year.

5. Given a positive number, print its square and square root. 6. The arithmetic mean of two numbers is the result of dividing their some by 2. The geometric mean of two numbers is the square root of their product. The harmonic mean of two numbers is the arithmetic mean of their reciprocals. Write a program that takes two floating-point numbers as inputs and displays these three means. 7. Write a C++ program to take a depth (in kilometers) inside the earth as input data; compute and display the temperature at that depth in degrees Celsius and Fahrenheit. The relevant formulas are:

Celsius = 10 x (depth) + 20 (Celsius temperature at depth in km) Farhrenheit = 1.8 x (Celsius) + 32

8. The Pythagorean Theorem states that the sum of the squares of the sides of a right triangle is equal to the square of the hypotenuse. For example, if two sides of a right triangle have lengths 3 and 4, then the hypotenuse must have a length of 5. The integers 3, 4, and 5 together form a Pythagorean triple. There is an infinite number of such triples. Given two positive integers, m and n, where m > n, a Pythagorean triple can be generated by the following formulas:

...The PythagoreanTheorem was one of the earliest theorems known to ancient civilizations. This famous theorem is named for the Greek mathematician and philosopher, Pythagoras. Pythagoras founded the Pythagorean School of Mathematics in Cortona, a Greek seaport in Southern Italy. He is credited with many contributions to mathematics although some of them may have actually been the work of his students.
ThePythagoreanTheorem is Pythagoras' most famous mathematical contribution. According to legend, Pythagoras was so happy when he discovered the theorem that he offered a sacrifice of oxen. The later discovery that the square root of 2 is irrational and therefore, cannot be expressed as a ratio of two integers, greatly troubled Pythagoras and his followers. They were devout in their belief that any two lengths were integral multiples of some unit length. Many attempts were made to suppress the knowledge that the square root of 2 is irrational. It is even said that the man who divulged the secret was drowned at sea.
The PythagoreanTheorem is a statement about triangles containing a right angle. The PythagoreanTheorem states that:
"The area of the square built upon the hypotenuse of a right triangle is equal to the sum of the areas of the squares upon the remaining sides."
Figure 1
According to...

...PythagoreanTheorem
Diana Lorance
MAT126
Dan Urbanski
March 3, 2013
PythagoreanTheorem
In this paper we are going to look at a problem that can be seen in the “Projects” section on page 620 of the Math in our World text. The problem discusses Pythagorean triples and asks if you can find more Pythagorean triples than the two that are listed which are (3,4, and 5) and (5,12, and 13) (Bluman, 2012). The Pythagoreantheorem states that for any right triangle, the sum of the squares of the length of the sides of the triangle is equal to the square of the length of the side opposite of the right angle (hypotenuse) and can be shown as a² + b² = c² (Bluman, 2012). We will be using a formula to find five more Pythagorean Triples and then verify each of them in the PythagoreanTheorem equation.
The formula that I have decided to illustrate is (2m)² + (m2 - 1)² = (m2 + 1)² where m is any natural number, this formula is attributed to Plato (c. 380 B. C.) (Edenfield, 1997). A natural number is any number starting from one that is not a fraction or negative (MathIsFun, 2011). The triples will be the square roots of each part of the equation. We will test this formula with the natural numbers of 5, 8, and 10. When we use 5 the formula looks like this: (2x5)² + (5² - 1)² = (5² +...

...In mathematics, the Pythagoreantheorem — or Pythagoras' theorem — is a relation in Euclidean geometry among the three sides of a right triangle (right-angled triangle). In terms of areas, it states:
In any right-angled triangle, 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 whose sides are the two legs (the two sides that meet at a right angle).
Thetheorem can be written as an equation relating the lengths of the sides a, b and c, often called the Pythagorean equation:[1]
where c represents the length of the hypotenuse, and a and b represent the lengths of the other two sides.
The Pythagoreantheorem is named after the Greek mathematician Pythagoras (ca. 570 BC—ca. 495 BC), who by tradition is credited with its discovery and proof,[2][3] although it is often argued that knowledge of the theorem predates him. There is evidence that Babylonian mathematicians understood the formula, although there is little surviving evidence that they used it in a mathematical framework.[4][5]
The theorem has numerous proofs, possibly the most of any mathematical theorem. These are very diverse, including both geometric proofs and algebraic proofs, with some dating back thousands of years. The theorem can be generalized...

...PYTHAGOREANTHEOREM
More than 4000 years ago, the Babyloneans and the Chinese already knew that a triangle with the sides of 3, 4 and 5 must be a right triangle. They used this knowledge to construct right angles. By dividing a string into twelve equal pieces and then laying it into a triangle so that one side is three, the second side four and the last side five sections long, they could easily construct a right angle.
A Greek scholar named Pythagoras, who lived around 500 BC, was also fascinated by triangles with these special side ratios. He studied them a bit closer and found that the two shorter sides of the triangles squared and then added together, equal exactly the square of the longest side. And he proved that this doesn't only work for the special triangles, but for any right triangle. Today we would write it somehow like this: a2 + b2= c2. In the time of Pythagoras they didn't use letters yet to replace variables. (They weren't introduced until the 16th century by Vieta.) Instead they wrote down everything in words, like this: if you have a right triangle, the squares of the two sides adjacent to the right angle will always be equal to the square of the longest side.
We can't be sure if Pythagoras really was the first person to have found this relationship between the sides of right triangles, since no texts written by him were found. In fact, we can't even prove the guy lived. But the theorem a2 + b2= c2 got his...

...THE WIZARD OF OZ 2
The Wizard of Oz
Scarecrow’s Speech on PythagoreanTheorem
The Pythagoreantheorem is one of the earliest theorems known to ancient civilization. The well-known theorem is named after the Greek mathematician and philosopher, Pythagoras. In the Wizard of Oz, after the Scarecrow gets a brain, he states the Pythagoreantheorem. However, he mistakenly says it applies to an isosceles triangle when it applies to a right triangle. He not only says the wrong triangle, he also gets the equation wrong.
The Scarecrow says, “The sum of the square root of two sides of an isosceles triangle is equal to the square root of the third side.” The correct equation for the Pythagoreantheorem is, “The sum of the squares of the two legs of a right triangle is equal to the square of the hypotenuse.” The isosceles triangle is a triangle with at least two equal sides; it also has two equal angles.
The Pythagoreantheorem is a statement about triangles containing a right angle. A right triangle is a triangle with a ninety-degree angle. With the Pythagoreantheorem, you take a triangle with a right angle and make a square on each of the three sides; the biggest square has the exact same area as the two other squares put together.
A square root of...

...functions of the angles. The basis for mensuration of triangles is the right-
angled triangle. The term trigonometry means literally the measurement of
triangles. Trigonometry is a branch of mathematics that developed from simple
measurements.
A theorem is the most important result in all of elementary mathematics. It was
the motivation for a wealth of advanced mathematics, such as Fermat's Last
Theorem and the theory of Hilbert space. ThePythagoreanTheorem asserts that
for a right triangle, the square of the hypotenuse is equal to the sum of the
squares of the other two sides. There are many ways to prove the PythagoreanTheorem. A particularly simple one is the scaling relationship for areas of
similar figures.
Did Pythagoras derive the PythagoreanTheorem or did he piece it together by
studying ancient cultures; Egypt, Mesopotamia, India and China? What did these
ancient cultures know about the theorem? Where was the theorem used in their
societies? In "Geometry and Algebra in Ancient Civilizations", the author
discusses who originally derived the PythagoreanTheorem. He quotes Proclos, a
commentator of Euclid's elements, "if we listen to those who wish to recount the
ancient history we may find some who refer this theorem to Pythagoras, and say
that he sacrificed an ox in...

...PythagoreanTheorem:
Some False Proofs
Even smart people make mistakes. Some mistakes are getting published and thus live for posterity to learn from. I'll list below some fallacious proofs of the Pythagoreantheorem that I came across. Some times the errors are subtle and involve circular reasoning or fact misinterpretation. On occasion, a glaring error is committed in logic and leaves one wondering how it could have avoided being noticed by the authors and editors.
Proof 1
One such error appears in the proof X of the collection by B. F. Yanney and J. A. Calderhead (Am Math Monthly, v.3, n. 6/7 (1896), 169-171.)
Suppose the theorem true. Then AB² = AC² + BC², BC² = CD² + BD², and AC² = AD² + CD². Combining the three we get
AB² = AD² + 2CD² + BD².
But CD² = AD·BD. Therefore,
AB² = AD² + 2AD·BD + BD².
From which
AB = AD + BD,
which is true. The supposition is true.
Critique
By the same token, assume 1 = 2. Then, by symmetry, 2 = 1. By Euclid's Second Common Notion, we may add the the two identities side by side: 3 = 3. Which is true, but does not make the assumption(1 = 2) even one bit less false.
As we know, falsity implies anything, truth in particular.
Proof 2
This proof is by E. S. Loomis (Am Math Monthly, v. 8, n. 11 (1901), 233.)
Let ABC be a right triangle whose sides are tangent to the circle O. Since CD = CF, BE = BF, and AE = AD = r = radius of circle, it is easily shown...

...In mathematics, the PythagoreanTheorem — or Pythagoras' theorem — is a relation in Euclidean geometry among the three sides of a right triangle (right-angled triangle). In terms of areas, it states:
In any right-angled triangle, 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 whose sides are the two legs (the two sides that meet at a right angle).
Thetheorem can be written as an equation relating the lengths of the sides a, b and c, often called the Pythagorean equation:
Where c represents the length of the hypotenuse, and a and b represent the lengths of the other two sides.
The PythagoreanTheorem is named after the Greek mathematician Pythagoras (ca. 570 BC—ca. 495 BC), who by tradition is credited with its discovery and proof.
The converse of the theorem is also true:
For any three positive numbers a, b, and c such that a2 + b2 = c2, there exists a triangle with sides a, b and c, and every such triangle has a right angle between the sides of lengths a and b.
An alternative statement is:
For any triangle with sides a, b, c, if a2 + b2 = c2, then the angle between a and b measures 90°.
This converse also appears in Euclid's Elements (Book I, Proposition 48):
"If in a triangle the square on one of...

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