Introduction

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## Introduction

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**Introduction**Geometric figures can be graphed in the coordinate plane, as well as manipulated. However, before sliding and reflecting figures, the definitions of some important concepts must be discussed. Each of the manipulations that will be discussed will move points along a parallel line, a perpendicular line, or a circular arc. In this lesson, each of these paths and their components will be introduced. 5.1.1: Defining Terms**Key Concepts**A point is not something with dimension; a point is a “somewhere.” A point is an exact position or location in a given plane. In the coordinate plane, these locations are referred to with an ordered pair (x, y), which tells us where the point is horizontally and vertically. The symbol A (x, y) is used to represent point A at the location (x, y). 5.1.1: Defining Terms**Key Concepts, continued**A line requires two points to be defined. A line is the set of points between two reference points and the infinite number of points that continue beyond those two points in either direction. A line is infinite, without beginning or end. This is shown in the diagram below with the use of arrows. The symbol is used to represent line AB. 5.1.1: Defining Terms**Key Concepts, continued**You can find the linear distance between two points on a given line. Distance along a line is written as d(PQ) where P and Q are points on a line. Like a line, a ray is defined by two points; however, a ray has only one endpoint. The symbol is used to represent ray AB. 5.1.1: Defining Terms**Key Concepts, continued**Similarly, a line segment is also defined by two points, but both of those points are endpoints. A line segment can be measured because it has two endpoints and finite length. Line segments are used to form geometric figures. The symbol is used to represent line segment AB. 5.1.1: Defining Terms**Key Concepts, continued**An angle is formed where two line segments or rays share an endpoint, or where a line intersects with another line, ray, or line segment. The difference in direction of the parts is called the angle. Angles can be measured in degrees or radians. The symbol is use to represent angle A. A represents the vertex of the angle. Sometimes it is necessary to use three letters to avoid confusion. In the diagram below, can be used to represent the same angle, . Notice that A is the vertex of the angle and it will always be listed in between the points on the angle’s rays. 5.1.1: Defining Terms**Key Concepts, continued**An acute angle measures less than 90° but greater than 0°. An obtuse angle measures greater than 90° but less than 180°. A right angle measures exactly 90°. Two relationships between lines that will help us define transformations are parallel and perpendicular. Parallel lines are two lines that have unique points and never cross. If parallel lines share one point, then they will share every point; in other words, a line is parallel to itself. 5.1.1: Defining Terms**Key Concepts, continued**Perpendicular lines meet at a right angle (90°), creating four right angles. 5.1.1: Defining Terms**Key Concepts, continued**A circle is the set of points on a plane at a certain distance, or radius, from a single point, the center. Notice that a radius is a line segment. Therefore, if we draw any two radii of a circle, we create an angle where the two radii share a common endpoint, the center of the circle. 5.1.1: Defining Terms**Key Concepts, continued**Creating an angle inside a circle allows us to define a circular arc, the set of points along the circle between the endpoints of the radii that are not shared. The arc length, or distance along a circular arc,is dependent on the length of the radius and the angle that creates the arc—the greater the radius or angle, the longer the arc. 5.1.1: Defining Terms**Common Errors/Misconceptions**mislabeling angles or not including enough points to specify an angle misusing terms and notations finding the length of incorrect arcs 5.1.1: Defining Terms**Guided Practice**Example 4 Given the following: Are and parallel? Are and parallel? Explain. 5.1.1: Defining Terms**Guided Practice: Example 4, continued**and intersect at the same angle and . will never cross . Therefore, is parallel to . 5.1.1: Defining Terms**Guided Practice: Example 4, continued**and intersect at the same angle, but . As you move from Z to Yon , you move closer to, and will eventually intersect, . Therefore, is not parallel to . ✔ 5.1.1: Defining Terms**Guided Practice: Example 4, continued**5.1.1: Defining Terms**Guided Practice**Example 5 Refer to the figures below. Given , is the set of points with center B a circle? Given , is the set of points with center Y a circle? 5.1.1: Defining Terms**Guided Practice: Example 5, continued**The set of points with center B is a circle because all points are equidistant from the center, B. 5.1.1: Defining Terms**Guided Practice: Example 5, continued**The set of points with center Y is not a circle because the points vary in distance from the center, Y. ✔ 5.1.1: Defining Terms**Guided Practice: Example 5, continued**5.1.1: Defining Terms**Introduction**The word transform means “to change.” In geometry, a transformation changes the position, shape, or size of a figure on a coordinate plane. The original figure, called a preimage, is changed or moved, and the resulting figure is called an image. We will be focusing on three different transformations: translations, reflections, and rotations. These transformations are all examples of isometry, meaning the new image is congruent to the preimage.**In this lesson, we will learn to describe transformations as**functions on points in the coordinate plane. the potential inputs for a transformation function f in the coordinate plane will be a real number coordinate pair, (x, y), and each output will be a real number coordinate pair, f(x, y) the x and y values will change. Example: f(x, y) = (x + 1, y + 2) means for any ordered pair (x, y) add 1 to the x coordinate and add 2 to the y coordinate**Finally, transformations are generally applied to a set of**points such as a line, triangle, square or other figure. In geometry, these figures are described by points, P, rather than coordinates (x, y), and transformation functions are often given the letters R, S, or T We will see T(x, y) written T(P) or P', known as “P prime.” A transformation T on a point P is a function where T(P) is P'.**Example 1**Given the point P(5, 3) and T(x, y)=(x + 2, y + 2), what are the coordinates of T(P)?**Guided Practice: Example 1, continued**Identify the point given. We are givenP(5, 3). 5.1.2: Transformations As Functions**Guided Practice: Example 1, continued**Identify the transformation. We are given T(P)=(x + 2, y + 2). 5.1.2: Transformations As Functions**Guided Practice: Example 1, continued**Calculate the new coordinate. T(P)=(x + 2, y + 2) (5 + 2, 3 + 2) (7, 5) T(P) = (7, 5) ✔ 5.1.2: Transformations As Functions**We can also define transformations using a subscript**notation. T(x, y)=(x + 2, y - 1) can be defined as T2, -1 meaning add two to the x value and subtract one from the y value. Example 2 Plot the points A’, B’, C’ using the translation T2, -1 B B’ A A’ A(1, 2) => B(4, 4) => C(3, 0) => A’(3, 1) B’(6, 3) C’(5, -1) C C’**Guided Practice**Example 3 Given the transformation of a translation T5, –3, and the points P (–2, 1) and Q (4, 1), show that the transformation of a translation is isometric by calculating the distances, or lengths, of and . 5.1.2: Transformations As Functions**Guided Practice: Example 3, continued**Plot the points of the preimage. 5.1.2: Transformations As Functions**Guided Practice: Example 3, continued**Transform the points. T5, –3(x, y) = (x + 5, y – 3) 5.1.2: Transformations As Functions**Guided Practice: Example 3, continued**Plot the image points. 5.1.2: Transformations As Functions**Guided Practice: Example 3, continued**Calculate the distance, d, of each segment from the preimage and the image and compare them. Since the line segments are horizontal, count the number of units the segment spans to determine the distance. d(PQ) = 5 The distances of the segments are the same. The translation of the segment is isometric. ✔ 5.1.2: Transformations As Functions**Guided Practice: Example 3, continued**5.1.2: Transformations As Functions