Unit 9 Transformations Homework 1 Translations Answers
Introduction
Unit 9 of a math curriculum often covers transformations, which involve changing the position, shape, or size of a figure. One common type of transformation is translation, which involves moving a figure without changing its orientation. Homework 1 in this unit typically focuses on translations, and in this article, we will provide answers and explanations for the questions in this assignment.
Question 1: Translate the Point
In this question, students are usually given a point and a vector, and they are asked to find the coordinates of the translated point. The vector represents the direction and distance of the translation. To find the coordinates of the translated point, you can add the components of the vector to the coordinates of the original point. For example, if the original point is (2, -3) and the vector is <4, 1>, the translated point would be (2 + 4, -3 + 1), which simplifies to (6, -2).
Question 2: Translate the Triangle
This question typically involves translating a triangle using a given vector. To translate a figure, you can apply the same vector to each of its vertices. For instance, if the original triangle has vertices (1, 2), (3, 4), and (5, 6), and the vector is <2, -1>, you can add the components of the vector to each vertex to find the coordinates of the translated triangle. The translated triangle would have vertices (1 + 2, 2 - 1), (3 + 2, 4 - 1), and (5 + 2, 6 - 1), which simplify to (3, 1), (5, 3), and (7, 5), respectively.
Question 3: Identify the Translation Type
In this question, students are usually given two figures and asked to determine if they are related by a translation. To do this, you can examine the distance and direction between corresponding points in the two figures. If the distance between corresponding points is the same and the direction is consistent, then the figures are related by a translation. For example, if the points (1, 2) and (4, 5) in the first figure correspond to the points (3, 4) and (6, 7) in the second figure, and the distance and direction between these points are the same, then the figures are related by a translation.
Question 4: Determine the Image
This question often involves determining the image of a figure after a translation. To do this, you can apply the same vector to each vertex of the figure. For instance, if the original figure has vertices (2, 3), (4, 5), and (6, 7), and the vector is <3, -2>, you can add the components of the vector to each vertex to find the coordinates of the image. The image would have vertices (2 + 3, 3 - 2), (4 + 3, 5 - 2), and (6 + 3, 7 - 2), which simplify to (5, 1), (7, 3), and (9, 5), respectively.
Question 5: Translate the Quadrilateral
This question typically involves translating a quadrilateral using a given vector. The process is similar to translating a triangle. You can apply the vector to each vertex of the quadrilateral to find the coordinates of the translated figure. For example, if the original quadrilateral has vertices (-1, 0), (1, 0), (1, 2), and (-1, 2), and the vector is <-2, 1>, the translated quadrilateral would have vertices (-1 - 2, 0 + 1), (1 - 2, 0 + 1), (1 - 2, 2 + 1), and (-1 - 2, 2 + 1), which simplify to (-3, 1), (-1, 1), (-1, 3), and (-3, 3), respectively.
Question 6: Translate the Rectangle
In this question, students are usually asked to translate a rectangle using a given vector. To do this, you can apply the same vector to each vertex of the rectangle. For example, if the original rectangle has vertices (0, 0), (4, 0), (4, 2), and (0, 2), and the vector is <1, -2>, the translated rectangle would have vertices (0 + 1, 0 - 2), (4 + 1, 0 - 2), (4 + 1, 2 - 2), and (0 + 1, 2 - 2), which simplify to (1, -2), (5, -2), (5, 0), and (1, 0), respectively.
Question 7: Determine the Translation Vector
This question often involves finding the translation vector given the coordinates of two corresponding points in the original and translated figures. To find the translation vector, you can subtract the coordinates of the original point from the coordinates of the translated point. For example, if the original point is (2, 3) and the translated point is (5, 7), the translation vector would be (5 - 2, 7 - 3), which simplifies to (3, 4).
Question 8: Translate the Circle
In this question, students are usually asked to translate a circle using a given vector. To do this, you can apply the same vector to the center of the circle. For example, if the original circle has a center at (-1, 2) and the vector is <2, -3>, the translated circle would have a center at (-1 + 2, 2 - 3), which simplifies to (1, -1).
Question 9: Translate the Parallelogram
This question typically involves translating a parallelogram using a given vector. The process is similar to translating a rectangle or quadrilateral. You can apply the vector to each vertex of the parallelogram to find the coordinates of the translated figure. For instance, if the original parallelogram has vertices (-2, 0), (1, 0), (3, 3), and (0, 3), and the vector is <2, -1>, the translated parallelogram would have vertices (-2 + 2, 0 - 1), (1 + 2, 0 - 1), (3 + 2, 3 - 1), and (0 + 2, 3 - 1), which simplify to (0, -1), (3, -1), (5, 2), and (2, 2), respectively.
Question 10: Translate the Polygon
In this question, students are usually asked to translate a polygon using a given vector. To do this, you can apply the same vector to each vertex of the polygon. For example, if the original polygon has vertices (-3, 0), (-2, 2), (0, 3), (2, 2), and (3, 0), and the vector is <1, -1>, the translated polygon would have vertices (-3 + 1, 0 - 1), (-2 + 1, 2 - 1), (0 + 1, 3 - 1), (2 + 1, 2 - 1), and (3 + 1, 0 - 1), which simplify to (-2, -1), (-1, 1), (1, 2), (3, 1), and (4, -1), respectively.
Question 11: Identify the Translation Rule
This question often involves identifying the translation rule that maps one figure onto another. To do this, you can examine the coordinates of corresponding points in the two figures. The translation rule consists of the vector that connects corresponding points. For instance, if the points (1, 2) and (4, 5) in the first figure correspond to the points (3, 4) and (6, 7) in the second figure, the translation rule would be <2, 2>, as the vector connecting these points is <2, 2>.
Question 12: Translate the Line Segment
In this question, students are usually asked to translate a line segment using a given vector. To do this, you can apply the same vector to each endpoint of the line segment. For example, if the original line segment has endpoints (-1, 0) and (1, 2), and the vector is <2, -1>, the translated line segment would have endpoints (-1 + 2, 0 - 1) and (1 + 2, 2 - 1), which simplify to (1, -1) and (3, 1), respectively.
Question 13: Determine the Translation Distance
This question often involves finding the distance of a translation given the coordinates of two corresponding points. To find the translation distance, you can calculate the distance between the original and translated points using the distance formula.