PowerPoint Presentation

[Virtual Presenter] A translation is a type of transformation that involves moving every point of a figure the same distance and direction. The size and shape of the figure remain unchanged during this process. In essence, a translation preserves the length and angle measures of a figure. For example, if we translate a triangle by two units along the x-axis, it will retain its original angles and side lengths. Similarly, if we translate a rectangle by three units along the y-axis, it retains its original area and perimeter. These characteristics define what makes a translation a rigid motion. Translations involve moving a figure from one location to another while keeping its size and shape intact. They are essential for identifying sequences of transformations that map a figure onto itself or onto another congruent or similar figure. Understanding the properties of translations enables us to analyze and apply transformations in various contexts..

[Audio] The original text has been rewritten in full sentences only, removing greetings, introductions, and thank-you statements. Here's the revised text: To begin drawing ABCD on a grid, use a pencil to sketch the shape. Using another color, draw a copy of ABCD on the grid in a different location with the same orientation. Label this new copy as QRST. Next, follow the instructions provided to move ABCD to the location of QRST. This may involve moving the shape horizontally or vertically. Once moved, compare your drawings of ABCD and QRST to see if they appear identical. If they do, then the instructions were followed correctly. To further test the instructions, exchange them with a partner and have each person attempt to recreate the third shape EFGH in another color on the same grid. By comparing the resulting drawings, you can determine whether the instructions were clear and effective. A good set of instructions should provide enough detail for anyone to understand them. Concise instructions should avoid unnecessary words and focus on the essential steps required to complete the task. Specific instructions should include precise language and measurements to ensure accuracy. By incorporating these elements, a set of instructions can effectively guide individuals through complex tasks like this explore-and-reason activity..

[Audio] ## Step 1: Define what a translation in geometry is. A translation in geometry is a transformation in a plane that moves every point of a figure the same distance and in the same direction. ## Step 2: Explain how a translation affects a figure. This means that the shape remains unchanged but its position changes. ## Step 3: Provide an example of a translation. In other words, it is a movement of a figure from one place to another without changing its size or shape. ## Step 4: Describe the formula used to represent a translation. The translation of a triangle, for instance, can be described using the formula T〈x, y〉 (△ABC) = △AʹBʹCʹ. ## Step 5: Explain the effect of applying a translation on a triangle. Here, x represents the horizontal shift and y represents the vertical shift. ## Step 6: Discuss the properties of translations. So, if we apply this translation to a triangle, its length and angles remain the same, but its position changes. ## Step 7: Classify translations as a type of rigid motion. This property makes translations a type of rigid motion. The final answer is:.

[Audio] The image of a translation is found by applying the translation to each vertex of the original figure. In this example, we have a triangle △EFG with vertices E(−5, 4), F(−1, 5), and G(−2, −1). We are asked to find the image of this triangle under the translation T〈7, −4〉. To do this, we add 7 to the x-coordinate and subtract 4 from the y-coordinate of each vertex. This gives us the new coordinates: Eʹ(2, 0), Fʹ(6, 1), and Gʹ(5, −5). These are the vertices of the image triangle △EʹFʹGʹ. Note that the translation does not simply map the vertices of the original triangle to the vertices of the image triangle, but rather it maps the entire original triangle onto the image triangle. Therefore, the resulting triangle △EʹFʹGʹ has the same size and shape as the original triangle △EFG..

[Audio] The translation T〈6, −7〉 moves each point of △EFG by 6 units to the right and 7 units down. The translation T〈11, 2〉 moves each point of △EFG by 11 units to the right and 2 units up. Applying these translations to the vertices of △EFG results in new coordinates for each vertex. For example, the vertex E(−5, 4) becomes Eʹ(1, −3) after applying the translation T〈6, −7〉. Similarly, the vertex F(−1, 5) becomes Fʹ(5, −2) after applying the translation T〈6, −7〉. The vertex G(−2, −1) becomes Gʹ(4, −8) after applying the translation T〈6, −7〉. After applying the translation T〈11, 2〉, the vertex E(−5, 4) becomes Eʹ(6, 6). The vertex F(−1, 5) becomes Fʹ(10, 7). The vertex G(−2, −1) becomes Gʹ(9, 1). The resulting coordinates of the vertices of △EʹFʹGʹ are therefore Eʹ(1, −3), Fʹ(5, −2), and Gʹ(4, −8) for the translation T〈6, −7〉, and Eʹ(6, 6), Fʹ(10, 7), and Gʹ(9, 1) for the translation T〈11, 2〉..

[Audio] The translation rule is given as T〈−1, −4〉, where each point is moved 1 unit to the right and 4 units down. The translation rule can also be expressed as T〈x + 1, y – 4〉, where each point is moved one unit to the right along the x-axis and four units downwards along the y-axis. Both expressions are equivalent and represent the same translation rule..

[Audio] The translation rule is T<5, 2>(P), which means that each point in the preimage will be moved 5 units to the right and 2 units up to get to the image. This rule applies to all points in the preimage, resulting in the image having the same shape and size as the preimage..

[Audio] The composition of rigid motions is a fundamental concept in geometry. It involves combining two rigid motions to create a new motion. The second motion is applied to the result of the first motion. This means that the final position of a figure is determined by applying both motions sequentially. For example, if we have a triangle and we apply a rotation followed by a reflection, the final position of the triangle is obtained by first rotating it and then reflecting it across a line. This process results in a new position of the triangle that is different from the original position. By combining these two motions, we get a single transformation that maps the original figure onto a new figure. This concept is essential in understanding how geometric figures change when they undergo various types of transformations. The composition of rigid motions is a combination of two rigid motions. The second motion is applied to the result of the first motion. This means that the final position of a figure is determined by applying both motions sequentially. We can think of it as performing one motion first and then the second motion on the resulting image. For instance, if we have a triangle and we apply a rotation followed by a reflection, the final position of the triangle is obtained by first rotating it and then reflecting it across a line. This process results in a new position of the triangle that is different from the original position. By combining these two motions, we get a single transformation that maps the original figure onto a new figure. This concept is essential in understanding how geometric figures change when they undergo various types of transformations. A composition of rigid motions is a combination of two rigid motions. The second motion is applied to the result of the first motion. This means that the final position of a figure is determined by applying both motions sequentially. If we have a triangle and we apply a rotation followed by a reflection, the final position of the triangle is obtained by first rotating it and then reflecting it across a line. This process results in a new position of the triangle that is different from the original position. Combining these two motions gives us a single transformation that maps the original figure onto a new figure. This concept is crucial for understanding how geometric figures change under various transformations..

[Audio] The translation T〈a, b〉 represents a change in coordinates by adding a units to the x-coordinate and b units to the translation. The translation T〈−a, −b〉 represents a change in coordinates by subtracting a units from the x-coordinate and b units from the translation. When composing two translations, we must apply the second translation to the result of the first translation. For example, if we have two translations T〈a, b〉 and T〈c, d〉, then the composition of these two translations is given by T〈a + c, b + d〉. In this case, we can use the composition formula: T〈a + c, b + d〉 = T〈a, b〉 ∘ T〈c, d〉. We will now apply this formula to find the composition of the two translations T〈2, 2〉 and T〈1, −1〉. First, we calculate the sum of the x-coordinates and the sum of the y-coordinates. The sum of the x-coordinates is 2 + 1 = 3, and the sum of the y-coordinates is 2 - 1 = 1. Therefore, the composition of the two translations is T〈3, 1〉. This translation adds 3 units to the x-coordinate and 1 unit to the y-coordinate, effectively moving an object 3 units to the right and 1 unit upwards..

[Audio] ## Step 1: Apply transformation T〈1, −1〉 to point 〈3, −2〉 Applying T〈1, −1〉 to point 〈3, −2〉 results in a new point 〈3+1, −2−1〉 = 〈4, −3〉. ## Step 2: Apply transformation T〈−2, 5〉 to the resulting point 〈4, −3〉 Applying T〈−2, 5〉 to point 〈4, −3〉 results in a new point 〈4−2, −3+5〉 = 〈2, 2〉. ## Step 3: Determine the composition of the given transformations The composition of the given transformations is T〈2, 2〉..

[Audio] The transformation that occurs when a triangle is reflected over a line parallel to one of its sides is called a glide reflection. Glide reflections are used to describe the transformations that occur when an object moves along a straight line while being reflected over a line parallel to that line. Glide reflections involve two steps: first, reflecting the object over a line parallel to one of its sides, and then translating the object by the length of that side. The result is a translation of the object by the length of the side, which is equivalent to moving the object along a straight line. This means that any object that undergoes a glide reflection will have the same shape and size as the original object, but with a different position. The key characteristic of a glide reflection is that it preserves the orientation of the object..

[Audio] Here is the rewritten text: The translation image that is equivalent to (rn ∘ rx-axis)(△DEF) can be determined by applying the concept of rn ∘ rx-axis to the given triangle, △DEF, with the vertices D(0, 0), E(0, 3), and F(3, 0)..

[Audio] The translation is a type of geometric transformation that involves moving a figure a certain distance in a specific direction. In this case, we have a translation that consists of two reflections across parallel lines. These reflections are perpendicular to the line connecting the preimage and image points. The key property of this translation is that the distance between the preimage and image is twice the distance between the two reflection lines. This means that if we apply the translation to a figure, it will move twice as far as the distance between the two reflection lines. We can see this in the example provided, where the translation moves the figure twice the distance between the two reflection lines. The proof shows us that this translation is equivalent to a single reflection across one of the parallel lines. So, we can conclude that a translation is indeed a composition of reflections across two parallel lines..

[Audio] The composition of two reflections across parallel lines is equivalent to a translation. This equivalence holds true for any point. The key to understanding this concept lies in visualizing the process of reflecting a point across two parallel lines. When we reflect a point across one line, we essentially flip the point over that line. Reflecting a point across two parallel lines involves flipping the point twice, which is equivalent to translating the point by the distance between the two lines. This is why the composition of two reflections across parallel lines is equivalent to a translation. The distance between the two lines determines the amount of translation. The more parallel the lines, the less translation required. As the lines become more divergent, the translation becomes greater. This relationship between the distance between parallel lines and the translation required is crucial to understanding the equivalence between the composition of reflections and translations. Understanding this relationship requires careful consideration of the geometric properties involved..

[Audio] The distance between Bʹ and n is d - x. The distance between n and B″ is also d - x. Therefore, the distance between Bʹ and B″ is equal to 2x. Since BB″ represents the distance between these two points, BB″ must be equal to 2d. The translation rn represents a reflection across the line n. The translation rm represents a reflection across the line m. These two lines are perpendicular to the line containing the preimage point and its corresponding image point. As a result, they are parallel to each other. We can express the translation as a composition of these two reflections. Specifically, if we denote the translation as T, then T(B) = (rn ∘ rm)(B). This composition of reflections achieves the same effect as the translation T. To justify why this proof is valid for any translation, we need to demonstrate that the choice of C and C″ was indeed arbitrary. Since the translation T maps C to C″, we can choose any point C and its corresponding image C″. Our proof holds true regardless of the specific values chosen for C and C″. In conclusion, the composition of reflections across parallel lines provides an alternative representation of a translation. By showing that this composition is equivalent to the original translation, we establish the validity of our proof for all possible translations..

[Audio] The process of proving a mathematical theorem involves several steps. First, one must identify the theorem to be proved. Next, one must understand the underlying assumptions and axioms that support the theorem. Then, one must develop a plan for the proof, including identifying key concepts and techniques that will be used. After that, one must apply these concepts and techniques to derive the desired result. Finally, one must verify that the proof is sound and free from errors. This process requires careful consideration of all possible cases and scenarios. A good proof should be clear, concise, and well-organized. A mathematician's ability to reason ly and logically is essential for creating effective proofs. The ability to communicate complex ideas clearly is also crucial..

[Audio] The first step in creating a new character is to decide on their personality traits, which can be influenced by their background, culture, and experiences. This includes determining their motivations, goals, and values. These characteristics will shape how they interact with others and make decisions throughout the story. The second step is to develop their physical appearance, including their height, weight, hair color, eye color, and any notable features. Their physical attributes should also reflect their personality traits. For example, if a character is introverted and shy, they may have a more reserved physical appearance. Similarly, if a character is confident and outgoing, they may have a more vibrant and energetic appearance. The third step is to create their backstory, including their family history, education, and significant life events. This information will help inform their personality traits and physical appearance. Finally, the fourth step is to determine their relationships with other characters, including friends, family members, and romantic partners. This will help create a sense of community and social dynamics within the story. By following these steps, you can create a well-rounded and believable character." Here is the rewritten text:.

[Audio] The translation can be visualized by imagining a rectangle with its sides parallel to the x-axis and y-axis. If we reflect the rectangle across both axes, we get a new rectangle with the same dimensions as the original one. This process is represented mathematically as T(△ABCD) = (rn ∘ rm)(△A′B′C′D′). Here, rn represents a reflection across the x-axis and rm represents a reflection across the y-axis. The resulting rectangle, △A′B′C′D′, has the same side lengths as the original rectangle, △ABCD, since AA′ = BB′ = CC′ = DD′ = 2d. This demonstrates that a translation preserves the length and angle measures of the original figure..

[Audio] The translation of a geometric figure is a transformation that moves every point of the figure by the same distance and in the same direction. The size and shape of the figure remain unchanged, but its position changes. A translation preserves the length and angle measures of a figure. A translation of a triangle along the x-axis moves it two units to the right. A translation along the y-axis moves it seven units upwards. When combining these translations, we get a new translation that moves the triangle both two units to the right and seven units upwards. This is equivalent to reflecting over the y-axis and then reflecting over the x-axis. Reflecting over the y-axis flips the triangle horizontally. Reflecting over the x-axis flips it vertically. Combining these reflections gives a net effect of moving the triangle two units to the right and seven units upwards. Therefore, reflecting over the y-axis and then reflecting over the x-axis is equivalent to translating the triangle two units to the right and seven units upwards. Now, let's examine the vocabulary section. We need to write an example of a composition of rigid motions for a given triangle. One way to do this is to start with a simple translation and then add one or more additional transformations. For instance, if we translate a triangle two units to the right and three units upwards, and then reflect it over the x-axis, we get a new triangle that is two units to the right, three units upwards, and flipped horizontally. We could also add a rotation to our initial translation. For example, if we rotate the triangle 90 degrees clockwise after translating it two units to the right and three units upwards, we get a new triangle that is two units to the right, three units upwards, and rotated 90 degrees clockwise. These examples illustrate how we can use different combinations of transformations to create new triangles. To find the values of x and y if T〈−2, 7〉 (x, y) = (3, −1), we need to think about what happens to the coordinates of the triangle under this translation. Since the translation moves the triangle two units to the right and seven units downwards, the new coordinates of the triangle will be x = 3 + 2 and y = -1 - 7. Solving these equations gives us x = 5 and y = -8. Therefore, the values of x and y are 5 and -8, respectively..

[Audio] ## Step 1: Understanding the problem To find the image of a translation, we need to apply the given translation to each vertex of the original triangle. ## Step 2: Applying the first translation Let's start with the first translation, T〈−4, −3〉 (△XYZ). To find the image of this translation, we need to add -4 to the x-coordinate and subtract 3 from the y-coordinate of each vertex. ## Step 3: Finding the image of vertex X For vertex X(1, −4), the new coordinates would be X'(1 - 4, -4 - 3) = X'(-3, -7). ## Step 4: Finding the image of vertex Y For vertex Y(-2, -1), the new coordinates would be Y'(-2 - 4, -1 - 3) = Y'(-6, -4). ## Step 5: Finding the image of vertex Z For vertex Z(3, 1), the new coordinates would be Z(3 - 4, 1 - 3) = Z(-1, -2). ## Step 6: Considering the second translation Next, let's consider the second translation, T〈5, −3〉 (△XYZ). We can apply the same process as before to find the image of this translation. ## Step 7: Finding the image of vertex X under the second translation For vertex X(1, −4), the new coordinates would be X'(1 + 5, -4 - 3) = X'(6, -7). ## Step 8: Finding the image of vertex Y under the second translation For vertex Y(-2, -1), the new coordinates would be Y'(-2 + 5, -1 - 3) = Y'(3, -4). ## Step 9: Finding the image of vertex Z under the second translation For vertex Z(3, 1), the new coordinates would be Z(3 + 5, 1 - 3) = Z(8, -2). ## Step 10: Determining the rule for the third translation Now, let's consider the third translation, which asks us to determine the rule for the translation shown. Looking at the table provided earlier, we see that the translation moves each point 7 units to the right and 4 units down. ## Step 11: Writing the rule for the third translation Therefore, the rule for this translation is T(x, y) = (x + 7, y - 4). ## Step 12: Composing translations for exercise 8 Moving on to exercises 8 and 9, we need to write the composition of translations as one translation. For exercise 8, we have T〈7, 8〉 ∘ T〈−3, −4〉. ## Step 13: Finding the result of the composition Using the rule for the second translation, we get T〈-3, -4〉(x, y) = (-3 + 7, -4 - 8) = (4, -12). ## Step 14: Applying the first translation to the result Then, applying the first translation, we get T〈7, 8〉(4, -12) = (4 + 7, -12 - 8) = (11, -20)..