Introduction

Introduction Think about trying to move a drop of water across a flat surface. If you try to push the water droplet, it will smear, stretch, and transfer onto your finger. The water droplet, a liquid, is not rigid. Now think about moving a block of wood across the same flat surface. A block of wood is solid or rigid, meaning it maintains its shape and size when you move it. You can push the block and it will keep the same size and shape as it moves. In this lesson, we will examine rigid motions, which are transformations done to an object that maintain the object’s shape and size or its segment lengths and angle measures. 5.5.1: Describing Rigid Motions and Predicting the Effects

Key Concepts Rigid motions are transformations that don’t affect an object’s shape and size. This means that corresponding sides and corresponding angle measures are preserved. When angle measures and sides are preserved they are congruent, which means they have the same shape and size. The congruency symbol ( ) is used to show that two figures are congruent. 5.5.1: Describing Rigid Motions and Predicting the Effects

Key Concepts, continued The figure before the transformation is called the preimage. The figure after the transformation is the image. Corresponding sides are the sides of two figures that lie in the same position relative to the figure. In transformations, the corresponding sides are the preimage and image sides, so and are corresponding sides and so on. 5.5.1: Describing Rigid Motions and Predicting the Effects

Key Concepts, continued Corresponding angles are the angles of two figures that lie in the same position relative to the figure. In transformations, the corresponding vertices are the preimage and image vertices, so ∠A and ∠A′ are corresponding vertices and so on. Transformations that are rigid motions are translations, reflections, and rotations. Transformations that are not rigid motions are dilations, vertical stretches or compressions, and horizontal stretches or compressions. 5.5.1: Describing Rigid Motions and Predicting the Effects

Key Concepts, continued Translations A translation is sometimes called a slide. In a translation, the figure is moved horizontally and/or vertically. The orientation of the figure remains the same. Connecting the corresponding vertices of the preimage and image will result in a set of parallel lines. 5.5.1: Describing Rigid Motions and Predicting the Effects

Key Concepts, continued 5.5.1: Describing Rigid Motions and Predicting the Effects

Key Concepts, continued Reflections A reflection creates a mirror image of the original figure over a reflection line. A reflection line can pass through the figure, be on the figure, or be outside the figure. Reflections are sometimes called flips. The orientation of the figure is changed in a reflection. 5.5.1: Describing Rigid Motions and Predicting the Effects

Key Concepts, continued In a reflection, the corresponding vertices of the preimage and image are equidistant from the line of reflection, meaning the distance from each vertex to the line of reflection is the same. The line of reflection is the perpendicular bisector of the segments that connect the corresponding vertices of the preimage and the image. 5.5.1: Describing Rigid Motions and Predicting the Effects

Key Concepts, continued Rotations A rotation moves all points of a figure along a circular arc about a point. Rotations are sometimes called turns. In a rotation, the orientation is changed. The point of rotation can lie on, inside, or outside the figure, and is the fixed location that the object is turned around. The angle of rotation is the measure of the angle created by the preimage vertex to the point of rotation to the image vertex. All of these angles are congruent when a figure is rotated. 5.5.1: Describing Rigid Motions and Predicting the Effects

Key Concepts, continued Rotating a figure clockwise moves the figure in a circular arc about the point of rotation in the same direction that the hands move on a clock. Rotating a figure counterclockwise moves the figure in a circular arc about the point of rotation in the opposite direction that the hands move on a clock. 5.5.1: Describing Rigid Motions and Predicting the Effects

Common Errors/Misconceptions creating the angle of rotation in a clockwise direction instead of a counterclockwise direction and vice versa reflecting a figure about a line other than the one given mistaking a rotation for a reflection misidentifying a translation as a reflection or a rotation 5.5.1: Describing Rigid Motions and Predicting the Effects

Guided Practice Example 2 Describe the transformation that has taken place in the diagram to the right. 5.5.1: Describing Rigid Motions and Predicting the Effects

Guided Practice: Example 2, continued Examine the orientation of the figures to determine if the orientation has changed or stayed the same. Look at the sides of the figures and pick a reference point. A good reference point is the outer right angle of the figure. From this point, examine the position of the “arms” of the figure. 5.5.1: Describing Rigid Motions and Predicting the Effects

Guided Practice: Example 2, continued 5.5.1: Describing Rigid Motions and Predicting the Effects

Guided Practice: Example 2, continued The orientation of the figures has changed. In the preimage, the outer right angle is in the bottom right-hand corner of the figure, with the shorter arm extending upward. In the image, the outer right angle is on the top right-hand side of the figure, with the shorter arm extending down. Preimage Image 5.5.1: Describing Rigid Motions and Predicting the Effects

Guided Practice: Example 2, continued Also, compare the slopes of the segments at the end of the longer arm. The slope of the segment at the end of the arm is positive in the preimage, but in the image the slope of the corresponding arm is negative. Preimage Image 5.5.1: Describing Rigid Motions and Predicting the Effects

Guided Practice: Example 2, continued A similar reversal has occurred with the segment at the end of the shorter arm. In the preimage, the segment at the end of the shorter arm is negative, while in the image the slope is positive. Preimage Image 5.5.1: Describing Rigid Motions and Predicting the Effects

Guided Practice: Example 2, continued Determine the transformation that has taken place. Since the orientation has changed, the transformation is either a reflection or a rotation. Since the orientation of the image is the mirror image of the preimage, the transformation is a reflection. The figure has been flipped over a line. 5.5.1: Describing Rigid Motions and Predicting the Effects

Guided Practice: Example 2, continued Determine the line of reflection. Connect some of the corresponding vertices of the figure. Choose one of the segments you created and construct the perpendicular bisector of the segment. Verify that this is the perpendicular bisector for all segments joining the corresponding vertices. This is the line of reflection. The line of reflection for this figure is y = –1, as shown on the next slide. 5.5.1: Describing Rigid Motions and Predicting the Effects

Guided Practice: Example 2, continued ✔ 5.5.1: Describing Rigid Motions and Predicting the Effects

Guided Practice Example 4 Rotate the given figure 45º counterclockwise about the origin. 5.5.1: Describing Rigid Motions and Predicting the Effects

Guided Practice: Example 4, continued Create the angle of rotation for the first vertex. Connect vertex A and the origin with a line segment. Label the point of reflection R. Then, use a protractor to measure a 45˚ angle. Use the segment from vertex A to the point of rotation R as one side of the angle. Mark a point X at 45˚. Draw a ray extending out from point R, connecting R and X. Copy onto . Label the endpoint that leads away from the origin , as shown on the next slide. 5.5.1: Describing Rigid Motions and Predicting the Effects

Guided Practice: Example 4, continued Create the angle of rotation for the second vertex.Connect vertex B and the origin with a line segment. The point of reflection is still R. Then, use a protractor to measure a 45˚ angle. Use the segment from vertex B to the point of rotation R as one side of the angle. Mark a point Y at 45˚. Draw a ray extending out from point R, connecting R and Y. Copy onto . Label the endpoint that leads away from the origin , as shown on the next slide. 5.5.1: Describing Rigid Motions and Predicting the Effects

Guided Practice: Example 4, continued Create the angle of rotation for the third vertex.Connect vertex C and the origin with a line segment. The point of reflection is still R. Then, use a protractor to measure a 45˚ angle. Use the segment from vertex C to the point of rotation R as one side of the angle. Mark a point Z at 45˚. Draw a ray extending out from point R, connecting R and Z and continuing outward from Z. Copy onto . Label the endpoint that leads away from the origin , as shown on the next slide. 5.5.1: Describing Rigid Motions and Predicting the Effects

Guided Practice: Example 4, continued Connect the rotated points.Theconnected points , , and form the rotated figure, as shown on the next slide. 5.5.1: Describing Rigid Motions and Predicting the Effects

Guided Practice: Example 4, continued ✔ 5.5.1: Describing Rigid Motions and Predicting the Effects

Introduction

Introduction

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Introduction to introduction to introduction to … Optimization

INTRODUCTION/ INTRODUCTION

Introduction

INTRODUCTION

Introduction

Introduction