Problem 1 - Algebra 2

Question

Triangle \(A B C\) with vertices \(A(1,4), B(2,-5),\) and \(C(-6,-6)\) is translated 3 units right and 1 unit down. Write the translation matrix.

Step-by-Step Solution

Answer

Translation matrix: \( \begin{pmatrix} 3 \\ -1 \end{pmatrix} \).

Step 1: Understanding Translation

In a translation, every point of a shape moves a constant distance in a specified direction. If a triangle's vertices are translated 3 units to the right and 1 unit down, each point's x-coordinate increases by 3, and each point's y-coordinate decreases by 1.

Step 2: Translation Matrix Structure

The translation matrix for moving a point (x, y) by \( h \) units horizontally and \( k \) units vertically is represented by the vector \( \begin{pmatrix} h \ k \end{pmatrix} \). In this problem, \( h = 3 \) and \( k = -1 \).

Step 3: Construct the Translation Matrix

Using the identified values of \( h \) and \( k \), construct the translation matrix as follows: \[ T = \begin{pmatrix} 3 \ -1 \end{pmatrix} \].

Key Concepts

Triangle Translation

Translating a triangle on a coordinate plane involves shifting every vertex of the triangle by adding or subtracting specific values to their coordinates. This geometric transformation ensures that the shape maintains its size, orientation, and dimensions after translation.

Consider a triangle with vertices named, say, \( A, B, C \). If a translation is given, such as "3 units right and 1 unit down", this translates to:

• Each \( x \)-coordinate of the vertices is increased by 3.
• Each \( y \)-coordinate of the vertices is decreased by 1.

This means that every vertex of the triangle moves uniformly in the defined direction. The result is a new set of vertices that describe the same triangle, just shifted in the coordinate plane. A pivotal notion here is that the triangle's shape does not alter; it's a direct translation on the graph.

Coordinate Plane

The coordinate plane is a two-dimensional surface where points are located by pairs of numbers, usually \( (x, y) \). This system serves as the setting for many geometric transformations, including translations.

Each point in the plane can be visualized as an intersection of horizontal (x-axis) and vertical (y-axis) values. Here’s how it functions:

• The horizontal axis, or x-axis, represents horizontal movements.
• The vertical axis, or y-axis, represents vertical movements.

When a triangle is translated on the coordinate plane, you essentially adjust these coordinates according to specific rules, without altering the graph's framework. The beauty of using a coordinate plane is in its precision and ability to easily visualize geometric changes, such as translations, rotations, and reflections.

Translation Vector

A translation vector indicates how far and in what direction to move a point or shape on the coordinate plane. The vector is expressed in terms of its horizontal and vertical components, typically written as \( \begin{pmatrix} h \ k \end{pmatrix} \), where:

• \( h \) is the horizontal shift. Positive \( h \) moves right, and negative \( h \) moves left.
• \( k \) is the vertical shift. Positive \( k \) moves up, and negative \( k \) moves down.

For the given triangle translation of "3 units right and 1 unit down":

• The translation vector becomes \( \begin{pmatrix} 3 \ -1 \end{pmatrix} \).

This vector acts as a guiding tool, ensuring each point of a shape moves concurrently across the coordinate plane without altering the shape's original form. By understanding and utilizing translation vectors, one can accurately modify geometric figures, giving them a new position while maintaining their properties.