A ratio is a method of comparing two values by dividing one of them by the other. This means that the ratio of A to B is equal to A divided by B. When you say that two things are directly proportional to each other, it means that they both increase and decrease at the same time and by the same multiplying factor. For example, the circumference of a circle is directly proportional to the diameter of that circle. When two things are directly proportional to each other if you divide the value of one of them by the value of the other you will get some constant value. This constant value is often referred to as a proportionality constant. The value of the proportionality constant that is the relationship between the circumference of a circle and the diameter of that same circle is represented by the Greek letter pi (117). Tm the value for 7: is equal to the ratio of the circumference of a circle to the diameter of that circle. When we say that two geometric shapes are \"similar\" we mean that they have the same basic shape. By this I mean that when we say that two objects are similar it means they have the same number of sides and the same angles between corresponding sides. If you have two objects that are similar it does not mean that they are the same size. But when two objects are similar you should be able to take one of them and through some combination of rotation and/ or enlargement you should be able to it to match the size and shape of the other. When we write a mathematical formula that gives the relationship between two or more values, we are often expressing the relationships between objects that are similar. For example when we write the equation for the area (A) of a circle based upon the radius (r) of that circle it looks like: A 2 at r2. Since all circles are similar to all other circles the relationship between the area of a circle and the radius of that circle is a constant. It does not matter how large or small the circle is upon which you are working. Once you get this relationship for any circle it holds for all other circles. The area (A) of a square is always equal to the length of any side (s) multiplied by itself. W the area of a square is calculated as: A232 The area (A) of a triangle can always be found by saying that it is equal to half of the base (b) of the triangle multiplied by the height (h) of that triangle. W the area of a triangle is calculated as: A = 1b h 2 A rational number is one that can be represented as the ratio of two integer values. An irrational number is one that can't be represented as the ratio of two integer values. The square-root of 2 is an irrational number. The value represented by 11? is an irrational number. In the past some people have rounded off the value of 71? to 22/7 to make the calculations easier and it is not uncommon to see the square root of 2 rounded off to 1.414. These rounded values are only approximations of the corresponding irrational numbers. When we say that a circle is inscribed in a square, it means that the circle is as large as it can be without having any part of the circle actually being outside of the square. When a circle is inscribed in a square the circle and the square will touch at four different points with the square and circle sharing only those four points. When we say that a square is inscribed in a circle, it means that the square is as large as it can be without having any part of the square actually being outside of the circle. When a square is inscribed in a circle the square and the circle will touch at four different points with the circle and square sharing only those four points. In part A of the lab you have a square that is inscribed inside a circle that is inscribed inside a square. What is the ratio of the area of the large square (the one in blue) to the area of the small square (the one in green)? Note: you are not allowed to simply draw these shapes and get the ratio based upon measurements from your drawing. Your answer needs to be based upon some combination of algebra, geometry and trigonometry alone. It can't be based in any part upon measurements you make from objects with these shapes. When we say that a circle is inscribed in a triangle, it means that the circle is as large as it can be without having any part of the circle actually being outside of the triangle. When a circle is inscribed in a triangle the circle and the triangle will touch at three different points with the triangle and circle sharing only those three A A In part B of the lab you have a circle that is inscribed inside an equilateral triangle. What is the ratio of the area of the triangle to the area of the circle? Note: you are not allowed to simply draw these shapes and get the ratio based upon measurements from your drawing. Your answer needs to be based upon some combination of algebra, geometry and trigonometry alone. It can't be based in any part upon measurements you make from objects with these shapes. Your nal answer should not be rounded off and written as a decimal number. Grading The explanation for part A of the lab will be worth 60 points. The explanation for part B of the lab will be worth 40 points. You will need to type up and submit a pdf le that explains how you obtained your answers. Feel free to include diagrams to help you explain how you obtained your nal answers. If I can't understand your explanation, then your explanation will be considered incorrect. (So clear diagrams really help.) Note: any drawings or sketches that you make may be hand drawn, but the explanations must be typed. I do not accept hand-written work in general because of the problems that I have had in the past trying to read student handwriting