Problem 100 - Algebra and Trigonometry

Question

Refer to Problem \(99 .\) The points \((-3,0),(-1,-2),(3,1),\) and (1,3) are the vertices of a parallelogram \(A B C D\). (a) Find the vertices of a new parallelogram \(A^{\prime} B^{\prime} C^{\prime} D^{\prime}\) if \(A B C D\) is translated by \(\mathbf{v}=\langle 3,-2\rangle\) (b) Find the vertices of a new parallelogram \(A^{\prime} B^{\prime} C^{\prime} D^{\prime}\) if \(A B C D\) is translated by \(-\frac{1}{2} \mathbf{v}\)

Step-by-Step Solution

Answer

Vertices of the new parallelogram for part (a) are (0, -2), (2, -4), (6, -1), and (4, 1). For part (b) they are (-9/2, 1), (-5/2, -1), (3/2, 2), and (-1/2, 4).

Step 1: Understanding the Translation Vector

Identify the translation vector \(\mathbf{v}=\langle 3,-2\rangle\). This vector indicates that every point will move 3 units to the right and 2 units downward.

Step 2: Translate Each Vertex by \(\mathbf{v}\)

To find the coordinates of the new parallelogram \(A^{\prime} B^{\prime} C^{\prime}D^{\prime}\), add the vector \(\mathbf{v}\) to each vertex of \(A B C D\).\(\mathbf{A}\(-3,0\)\rightarrow \mathbf{A^{\prime}}(-3+3, 0-2) = (0, -2)\),\(\mathbf{B}\(-1,-2\)\rightarrow \mathbf{B^{\prime}}(-1+3, -2-2) = (2, -4)\),\(\mathbf{C}\(3,1\)\rightarrow \mathbf{C^{\prime}}(3+3, 1-2) = (6, -1)\),\(\mathbf{D}\(1,3\)\rightarrow \mathbf{D^{\prime}}(1+3, 3-2) = (4, 1)\).

Step 3: Finding the New Vertices with \(-\frac{1}{2} \mathbf{v}\)

To find the coordinates of the new parallelogram \(A^{\prime} B^{\prime} C^{\prime}D^{\prime}\) under translation by \(-\frac{1}{2} \mathbf{v}\), calculate \(-\frac{1}{2} \mathbf{v} = -\frac{1}{2} \langle 3, -2 \rangle = \langle -\frac{3}{2}, 1 \rangle\). Then add this new vector to each vertex.\(\mathbf{A}\(-3,0\)\rightarrow \mathbf{A^{\prime}}(-3 - \frac{3}{2}, 0 + 1) = (-\frac{9}{2}, 1)\),\(\mathbf{B}\(-1,-2\)\rightarrow \mathbf{B^{\prime}}(-1 - \frac{3}{2}, -2 + 1) = (-\frac{5}{2}, -1)\),\(\mathbf{C}\(3,1\)\rightarrow \mathbf{C^{\prime}}(3 - \frac{3}{2}, 1 + 1) = (\frac{3}{2}, 2)\),\(\mathbf{D}\(1,3\)\rightarrow \mathbf{D^{\prime}}(1 - \frac{3}{2}, 3 + 1) = (-\frac{1}{2}, 4)\).

Key Concepts

parallelogram vertices

When dealing with parallelogram vertices in coordinate geometry, it is essential to correctly identify the coordinates of each vertex. A vertex is simply a corner point of the geometric shape. In the given problem, we have a parallelogram named \(ABCD\) with its vertices at \(-3,0\), \(-1,-2\), \(3,1\), and \(1,3\). A key trait of parallelograms is that opposite sides are parallel and equal in length. By identifying these four points as vertices, we can manipulate and translate the shape while maintaining its structural properties. To ensure clarity and prevent confusion, always double-check each vertex's coordinates before proceeding with any transformations. This step lays the groundwork for accurate translations and geometric operations.

translation vector

A translation vector provides the necessary information to move every point of a shape uniformly. In this problem, the translation vector is given as \(\textbf{v} = \langle 3, -2 \rangle \). This means each vertex of the parallelogram will move 3 units to the right and 2 units down. Translating by this vector involves adding the vector components directly to the coordinates of each vertex. Let’s take a closer look at how this operates for each vertex of the parallelogram:

• For vertex A(-3, 0), add the vector to get new coordinates: \( (-3 + 3, 0 - 2) = (0, -2) \).
• Similarly, for vertex B(-1, -2), you get: \( (-1 + 3, -2 - 2) = (2, -4) \).
• Continue this process for C(3, 1) and D(1, 3) to get new vertices C'(6, -1) and D'(4, 1).

By repeating this straightforward process, you can accurately translate entire shapes within coordinate geometry, preserving the original geometric properties.

coordinate geometry

Coordinate geometry involves plotting and analyzing geometric shapes in a coordinate system, usually the Cartesian plane. Each point is defined by an \(x\) and \(y\) coordinate. In this exercise, we apply coordinate geometry principles to translate a parallelogram. Key steps in solving the problem include:

• Identifying the coordinates of the vertices: \(-3,0\), \(-1,-2\), \(3,1\), and \(1,3\).
• Applying the translation vector \(\textbf{v} = \langle 3, -2 \rangle \), to shift the parallelogram's position.
• Breaking down the vector operations into simple addition/subtraction steps for each coordinate.
• Recognizing that the translated shape retains its parallel and equal sides, reflecting the properties of a parallelogram.

Understanding coordinate geometry is crucial for visualizing and manipulating shapes, ensuring precise transformations and maintaining geometric integrity.