# Pascal's triangle

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Introduction I’m going to start this paper with introducing the history of Pascal’s Triangle, time ranging from 2nd century BC to 18th century AD. Among all the academic works by ancient mathematicians, I’m going to focus on Jiu Zhang Suan Shu [Nine Chapters on the Mathematical Art] composed by Chinese scholars around 2nd century BC-10th century AD. I will use the binomial expansions derived from Pascal’s triangle as an example and then illustrate the coefficients expansion in general. After that, I will introduce Jade Mirror of the Four Unknowns written by Zhu Shijie around 14th century AD. My focus will be his idea of using the “four unknowns” to convert a problem into a mathematical system of polynomial equations. Later on, I will also…

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different types of shaped triangles you may find in mathematics. They have learned many of the basic shapes of triangles, for example, the right triangle, acute triangle, equilateral triangle, and lastly the isosceles triangle. As a fun way to incorporate what the students have learned thus far in the lesson, I am going to have them construct toothpick bridges solely out of toothpicks and mini marshmallows. Why is this lesson important: This lesson allows for my students to be creative by…

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These things don't exist in math. Math is full of "rules" that don't have exceptions. Things such as the Pythagorean Theorem will always work for right triangles. There will never be a right triangle where a^2 + b^2 does not = c^2. This is what I like about math. Math is a subject that I think would never Let me down ; It has no contradictions. It's something that's reliable and useful. I've learned that without math, life would be a cycle of events without reason. If I ever wonder why when I…

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At about twenty centuries ago there was an amazing discovery about right angled triangles: “In a right angled triangle the square of the hypotenuse is equal to the sum of squares of the other two sides.” It is called Pythagoras Theorem and can be written in one short equation: a²+b²=c² where c is the longest side of triangle and a and b are the other two sides. Pythagoras was born in the island of Samos in 570 BC in Greek in the eastern Agean. He was the son of Mnesarchus and his mother's name…

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We can easily convert this into Cartesian form by replacing z by x+iy |(x+iy)-(-3 +2i)|=5 |(x+3) + i(y-2)|=5 √(〖(x+3)〗^2+〖(y-2)〗^2 ) =5 or 〖(x+3)〗^2+〖(y-2)〗^2=25 Which is the exact form of a Cartesian equation of a circle with a center of (-3, 2) and a radius of 5 We could check if a particular point lies on the circle Ex. z2=1-3i Substituting into the equation of the circle, with z replaced by z2=1-3i |(1-3i) - (-3 + 2i)|=5 |4-5i|=5 √(4^2+〖(-5)〗^2 ) =5 41=5, which is not true, thus z2 does not…

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tree sides of a right triangle. It’s the sum of the areas of the two squares on the legs equals the area of the square on the hypotenuse. The equation use for it is A squared plus B squared equals C squared.The Theorem relates the lengths of the three sides of any right triangle. The theorem is named after the ancient Greek. There is evidence that indicates that Pythagorean Theorem was well- known to the mathematicians of the first Babylonian Dynasty (20th to 16th centuries BC) which would…

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At first, as I was examining the image of the swimmer, I thought about determining the area between the swimmer’s legs. At first, I thought that I could relate it to the area of a triangle, but later acquired the fact that the side length and height of an isosceles triangle cannot be equivalent, which meant that the area of a triangle would not fit this image. Figure 2 Created in Microsoft Paint In Figure 2, I have attempted to display an isosceles triangle to put in place of the swimmer's…

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properties of right-angled triangles seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry, and it was touched on in some of the most ancient mathematical texts from Egypt , dating from over a thousand years earlier. One of the simplest proofs comes from ancient China, and probably dates from well before Pythagoras' birth. It was Pythagoras, though, who gave the theorem its definitive form, although it is not clear whether Pythagoras himself…

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I'm not exactly sure how to solve this question I might have forgot how to do it but I'm going to guess that based off the picture & what I know about circles I can say that all circles if this makes sense are always similar to each other & that's always no matter what the radius might be ' The steps are as follows... * Obviously draw a triangle * Insert angle bisector for each angle of the tri. Insert the lines & make sure that your long to the point where you can see the point of…

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School to develop our wide spectrum of ideas and abilities into a highly sophisticated product. While the studies at Governor’s School are noticeably more advanced and require more effort than at regular public schools, I see this rigor as the key to my academic success. For me, the classes I take that constantly introduce new thoughts that test my capability to “think outside the box”, are the ones that capture all my attention and interest. For example, while working with the Sierpinski…

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