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Throughout Ch. 9, I improved greatly along the way. I started off weak, not paying attention to details such as negative signs and simplifying radicals. But as I continued to correct my homework, I realized that I needed to slow down when completing it and read problems all the way through. Also, Mrs. Kudlov began commenting on my homework, and I realized that I needed to draw diagrams. This has made a TREMENDOUS difference in the accuracy of my homework. I am able to see things that I was not able to before when I was not drawing diagrams, which was previously my weakness in Ch. 9. Now that I have begun to draw diagrams and write down given information in my notes, it has also impacted my quizzes and helped me perform well on them. I was not pleased with how I performed on my first quiz, so I decided to retake it. This was a smart decision because I achieved much improvement. I think this is mostly because of the fact that I began do a better job on my homework. For future chapters, I want to continue to correct my homework, draw diagrams, and pay attention to detail. Also, I will continue to come in early and go to math lab, which contributed to my success this chapter. I find it useful to get help from many different sources because they each bring a different approach to the problem. This ultimately helps me during quizzes when I am able to use of the approaches. This Padlet has helped me review not only the formulas for right triangles, but also the basic facts that everything is built off of for right triangles. I feel prepared to take my Ch. 9 test after completing this Padlet.

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9.8 Pythagorean Theorem with Solid Figures

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9.8 Pythagorean Theorem with Solid Figures

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This image from the 9.8 notes shows how the Pythagorean Theorem can be applied to solid figures, usually in two step processes. Rectangular solids and regular square pyramids are often used. The diagonals of both faces and opposite planes must be found, as well as things such as slant height, altitude, and perimeter. Like I mentioned before, it is crucial to know Pythagorean Theorem because it is applied in situations like this. In order to visualize problems with solid figures, it is best to draw them. These examples clearly highlight how to solve for certain segments or lengths.

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This picture from the 9.7 notes gives the ratio of the sides of a 45-45-90 triangle as x: x: x root 2. This is useful because it is easily recognizable in right isosceles triangles and make solving right triangle problems with them much easier and quicker. Like the chart used for 30-60-90 triangles, they same can be used for the ratio of sides of a 45-45-90 triangle.

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This image of our 9.7 notes clearly shows that the ratio of the sides of a 30-60-90 triangle are x: x root 3: 2x. This, like the Pythagorean Triples, is very useful because it enables you to easily recognize and solve right triangle problems. In examples A and B, a helpful chart was used to organize the given information and what I needed to know. Using this chart helps see what the variables are and what I am looking to solve/how I can solve it.

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The principle of the reduced triangle is exhibited in two ways here. In the first example, there is a triangle smaller than the original Pythagorean Triple, and in the second example, the triangle is larger. This is possible due to the fact that all sides were increased in the same ratio. It is best to work with triangles in their reduced form because they are easy to memorize. If a side is missing, the other sides should be put in their reduced triangle form. Once the answer is found, all that has to be done is that it is multiplied by the same ratio it was increased or decreased.

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This is a chart from our 9.6 notes with a list of simplified and non-simplified families of Pythagorean Triples. These are extremely important to know because they make many equations much easier and quicker to compute. Also, when there are different versions of these reduced triangles, they are easy to recognize since we have memorized the Pythagorean triples.

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This example is good because it includes the distance formula and it shows how it is used in situations when the coordinates of two points are given. This can be applied in right triangles, because "c" always ends up being the same as the answer given from the distance formula. The coordinates of one point are given subscripts of 1 and the coordinates of the other point are given subscripts of 2. This helps to organize them when inserting them into the equation.

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9.4 Pythagorean Theorem and Its Converse

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9.4 Pythagorean Theorem and Its Converse

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This first example from our 9.4 notes is useful because it uses the Pythagorean Theorem and its converse. The Pythagorean Theorem states that a^2+b^2=c^2. If the first value is greater than the value of c squared, then the triangle is acute, and vice versa, as shown in the photograph below. The example below it, 2a, shows a situation where the Pythagorean Theorem is applicable- that is, when one side of a triangle is missing. This is essential to know because it may be used multiple times throughout one problem in order to get one final answer.

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This entire page from the 9.3 notes is extremely helpful in explaining the altitude/hypotenuse theorems. Not only does in include a diagram with the labeled parts of the equation, but it also includes the equation and the three different proportions that can be written. This is very useful because many concepts are later built on it and it is necessary to understand hypotenuse/altitude problems.

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9.1 Simplifying Radical Equations and Solving Quadratic Equations

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9.1 Simplifying Radical Equations and Solving Quadratic Equations

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This example includes division and multiplication of radicals. In order to get radicals out of the denominator, they must be multiplied to the equation in a ratio of 1:1. In this case, root 5 over root 3 had to be multiplied by root 3 over root 3. Problems like these are important because they show how you cannot have radicals in the denominator. Whenever there is a radical in the denominator, the equation is not simplified enough.