Problem 15 - Algebra 2

Question

Write the translation matrix for each figure. Then find the coordinates of the image after the translation. Graph the preimage and the image on a coordinate plane. Triangle \(D E F\) with vertices \(D(-2,2), E(3,5),\) and \(F(5,-2)\) is translated so that \(D^{\prime}\) is at \((1,-5) .\) Find the coordinates of \(E^{\prime}\) and \(F^{\prime} .\)

Step-by-Step Solution

Answer

The coordinates after translation are: \( E' (6,-2) \) and \( F' (8,-9) \).

Step 1: Determine the Translation Vector

First, we need to determine the translation vector that will move vertex \(D\) from \((-2,2)\) to its new position \((1,-5)\). To find this, we subtract the original coordinates of \(D\) from the coordinates of \(D'\):\[\begin{align*}\text{Translation Vector} &= (1 - (-2), -5 - 2) \&= (1 + 2, -5 - 2) \&= (3, -7).\end{align*}\]So, the translation vector is \((3,-7)\).

Step 2: Apply Translation to Vertex E

Using the translation vector \((3, -7)\), we find the new coordinates of \(E\) by adding this vector to the original coordinates of \(E\), which are \((3, 5)\):\[\begin{align*}E'(x', y') &= (3 + 3, 5 - 7) \&= (6, -2).\end{align*}\]Thus, the coordinates of \(E'\) are \((6, -2)\).

Step 3: Apply Translation to Vertex F

Similarly, apply the translation vector \((3, -7)\) to the coordinates of \(F\), which are \((5, -2)\):\[\begin{align*}F'(x', y') &= (5 + 3, -2 - 7) \&= (8, -9).\end{align*}\]Hence, the coordinates of \(F'\) are \((8, -9)\).

Step 4: Summarize Coordinates of Images and Graph

Now we have the coordinates of the image after translation: \(D' = (1,-5)\), \(E' = (6,-2)\), and \(F' = (8,-9)\). These points should be plotted on a coordinate plane alongside the original points \(D (-2, 2)\), \(E (3,5)\), and \(F (5,-2)\) to visualize the translation.

Key Concepts

Coordinate Plane

The coordinate plane is a two-dimensional space defined by a horizontal axis called the x-axis and a vertical axis called the y-axis. Any point on this plane is identified by a pair of numbers known as coordinates, written as \(x, y\). The coordinate plane is essential for graphing geometric shapes and calculating transformations.

In our exercise, we're working with triangle \(DEF\), which has its vertices located at specific coordinates: \(D(-2, 2), E(3, 5),\) and \(F(5,-2)\). These coordinates help us determine the location of each vertex before any transformation occurs.

When graphing shapes or performing translations, the coordinate plane provides a visual representation and ensures precision in calculations. Understanding how to plot points accurately is fundamental to mastering transformations like translations.

Translation Vector

A translation vector is a mathematical tool used to move every point of a shape to a new location on the coordinate plane without altering its shape, size, or orientation. It is expressed as a pair of numbers, such as \( (3, -7) \), which tell us how far and in what direction to move the points.

For example, moving from \(D(-2, 2)\) to \(D'(1, -5)\) involves applying the translation vector \((3, -7)\). This vector represents a shift right by 3 units and down by 7 units.

When determining a translation vector, subtract the original coordinates from the new coordinates. If a point's new coordinate is \(1\) compared to its original \(-2\), it has moved \(1 - (-2) = 3\) units in the x-direction, making translations systematic and predictable.

Vertex Transformation

Vertex transformation occurs when a vertex (corner point) of a shape is moved using a translation vector. Each vertex undergoes this transformation separately, thus maintaining the geometrical properties of the shape.

In this lesson, vertex \(D\) is initially transformed from \( (-2, 2) \) to \( (1, -5) \), setting the stage for further transformations using the derived translation vector \( (3, -7) \).

We then apply this vector to vertex \( E \) as follows: adding 3 to the x-coordinate and subtracting 7 from the y-coordinate:

• Original coordinates: \(E(3, 5)\)
• New coordinates: \(E'(6, -2)\)

Perform a similar computation for vertex \( F \) to find \( F'(8, -9)\). Successfully transforming each vertex confirms the shape's identical size and orientation post-translation.

Graphing Translations

Graphing translations involves plotting both the original and the translated shapes on the coordinate plane to visually demonstrate the movement. This is particularly useful in reinforcing understanding and ensuring accuracy.

On your graph, begin by plotting the initial vertices of triangle \( DEF \): \( D(-2, 2)\), \( E(3, 5)\), and \( F(5, -2)\). Once the translation vector \( (3, -7) \) is applied, plot the new vertices: \( D'(1, -5) \), \( E'(6, -2) \), and \( F'(8, -9) \).

• Use straight lines to connect the vertices, forming triangles before and after translation.
• Observe the parallel shift of each point, maintaining the shape's uniformity.

This visual representation not only highlights the translation's effect but also reinforces the importance of accuracy in plotting and calculating translations by showing the congruence between the pre-image and the image.