Problem 40 - Algebra 2

Question

Triangle ABC with vertices A(0, 2), B(-3, -1), and C(-2, -4) is translated 1 unit right and 3 units up. Find the coordinates of \(\triangle A^{\prime} B^{\prime} C^{\prime} .\) Then graph the preimage and the image.

Step-by-Step Solution

Answer

The coordinates of \(\triangle A^{\prime} B^{\prime} C^{\prime}\) are A'(1, 5), B'(-2, 2), and C'(-1, -1).

Step 1: Understand the Translation

In this exercise, we are given a triangle with specific vertices, and we need to translate it. This involves shifting each point of the triangle 1 unit to the right and 3 units upwards.

Step 2: Translate Vertex A

The initial coordinates of point A are (0, 2). To translate this point, add 1 to the x-coordinate and 3 to the y-coordinate: \[ A^{\prime} = (0+1, 2+3) = (1, 5) \]

Step 3: Translate Vertex B

The initial coordinates of point B are (-3, -1). Add 1 to the x-coordinate and 3 to the y-coordinate: \[ B^{\prime} = (-3+1, -1+3) = (-2, 2) \]

Step 4: Translate Vertex C

The initial coordinates of point C are (-2, -4). Add 1 to the x-coordinate and 3 to the y-coordinate: \[ C^{\prime} = (-2+1, -4+3) = (-1, -1) \]

Step 5: Summary of Translated Coordinates

The vertices of the translated triangle \(\triangle A^{\prime} B^{\prime} C^{\prime}\) are: - \( A^{\prime} = (1, 5) \) - \( B^{\prime} = (-2, 2) \) - \( C^{\prime} = (-1, -1) \)

Step 6: Graph the Preimage and Image

To graph the preimage and the image:1. Plot the original points A(0, 2), B(-3, -1), and C(-2, -4) to visualize \(\triangle ABC\).2. Plot the translated points \( A^{\prime} (1, 5), B^{\prime} (-2, 2), C^{\prime} (-1, -1) \) to visualize \(\triangle A^{\prime} B^{\prime} C^{\prime}\).

Key Concepts

Translation

Translation in geometry refers to sliding each point of a shape a certain distance in a specified direction. It does not involve rotating or resizing the shape; it simply moves it. In the context of this exercise, translation is used to shift every vertex of triangle ABC by a fixed amount: 1 unit to the right and 3 units upwards.

To translate a point, you adjust its coordinates by adding to the x-coordinate and y-coordinate. For Triangle ABC, translation involves:

• Moving each x-coordinate 1 unit to the right, effectively adding 1.
• Moving each y-coordinate 3 units up, effectively adding 3.

This changes the entire position of the triangle on the coordinate plane without altering its shape or orientation.

Coordinate Geometry

The system of coordinate geometry helps us precisely locate points on a plane by using pairs of numbers called coordinates. Each point on a plane is represented by an ordered pair \((x, y)\). The coordinate geometry system underpins many mathematical operations, including translations.

For triangle ABC, each vertex is placed on a coordinate system, described in terms of its x (horizontal) and y (vertical) components. In our specific problem:

• The original vertices are A(0, 2), B(-3, -1), and C(-2, -4).
• After translation, the new vertices are \( A^{\prime} (1, 5), B^{\prime} (-2, 2), C^{\prime} (-1, -1) \).

Coordinate geometry allows us to visualize these points and transformations, providing a clear picture of the shape's movement across the plane.

Plotting Points

Plotting points is the action of marking specific coordinates on a graph, enabling a visual representation of geometric figures. This process helps us see both the original shapes and their transformations.

For this triangle exercise:

• First, plot the initial vertices of triangle ABC on the graph. This involves marking the points A(0, 2), B(-3, -1), and C(-2, -4).
• Then, mark the translated points \( A^{\prime} (1, 5), B^{\prime} (-2, 2), C^{\prime} (-1, -1) \) to see the new position of triangle \( A^{\prime} B^{\prime} C^{\prime} \).
• Comparing the original and transformed triangles on the graph provides a visual understanding of how translation has shifted the shape.

Plotting points is crucial for visual clarity and helps break down abstract mathematical concepts into tangible, understandable parts.