Problem 16 - Algebra 2

Question

Write the vertex matrix for each figure. Then find the coordinates of the image after the dilation. Graph the preimage and the image on a coordinate plane. \(\triangle A B C\) with \(A(0,2), B(1.5,-1.5),\) and \(C(-2.5,0)\) is dilated so that its perimeter is three times the original perimeter.

Step-by-Step Solution

Answer

Dilated image vertices: A'(0,6), B'(4.5,-4.5), C'(-7.5,0).

Step 1: Define the Vertex Matrix

First, represent the vertices of \( \triangle ABC \) in matrix form. The vertex matrix for the triangle is composed of the coordinates of points \( A \), \( B \), and \( C \) arranged in rows: \[\begin{bmatrix}0 & 2 \1.5 & -1.5 \-2.5 & 0\end{bmatrix}\]

Step 2: Calculate the Dilation Factor

The problem states that the perimeter of the image is three times the perimeter of the original triangle. Therefore, the dilation factor \( k \) must be 3. In other words, every side length of \( \triangle ABC \) will be multiplied by \( k = 3 \). This means each coordinate will also be multiplied by 3 since the figure is enlarged uniformly.

Step 3: Apply the Dilation to the Vertex Matrix

Multiply each coordinate in the vertex matrix by the dilation factor \( k = 3 \):The dilated vertex matrix will be: \[\begin{bmatrix}0 \times 3 & 2 \times 3 \1.5 \times 3 & -1.5 \times 3 \-2.5 \times 3 & 0 \times 3\end{bmatrix} = \begin{bmatrix}0 & 6 \4.5 & -4.5 \-7.5 & 0\end{bmatrix}\]

Step 4: Identify the Coordinates of the Image

The coordinates of the image \( \triangle A'B'C' \) after the dilation are:- \( A'(0, 6) \)- \( B'(4.5, -4.5) \)- \( C'(-7.5, 0) \).

Step 5: Graph the Preimage and the Image

On a coordinate plane, plot the original triangle using points \( A(0,2) \), \( B(1.5,-1.5) \), and \( C(-2.5,0) \). Then, plot the dilated triangle using points \( A'(0, 6) \), \( B'(4.5, -4.5) \), and \( C'(-7.5, 0) \). Draw both triangles to visualize the dilation, ensuring that the image is three times the original size in terms of perimeter.

Key Concepts

Vertex Matrix

When dealing with transformations in geometry, the vertex matrix plays a crucial role. Imagine you have a triangle with specific coordinates for its vertices. To perform calculations or transformations, it's helpful to organize these coordinates into a matrix form. This arrangement, called the vertex matrix, consists of rows where each vertex's coordinates are listed as pairs. For example, the vertex matrix of \( \triangle ABC \) is:\[\begin{bmatrix}0 & 2 \ 1.5 & -1.5 \ -2.5 & 0\end{bmatrix}\]This matrix acts as a foundation for further operations and allows for easier handling of the vertices in transformation equations.

Coordinate Geometry

Coordinate geometry, or analytic geometry, mobilizes algebra and geometry together to study objects like triangles on a coordinate plane. It gives us the power to manipulate the shape's vertices algebraically, which translates to geometric transformations.To do this, each vertex of a shape is assigned a coordinate. These coordinates act as fixed points on the grid, allowing us to perform precise transformations—like shifting, resizing, or rotating the shape.

• The x-coordinate tells you how far to move left or right.
• The y-coordinate tells you how far to move up or down.

In the example of \( \triangle ABC \), the coordinates before dilation are \((0,2), (1.5,-1.5), (-2.5,0)\). It's this toolbox of numbers that lets us see transformations vividly on paper or screens.

Triangle Transformation

Transformations involve changing a shape's position, size, or orientation. In terms of coordinate geometry, common transformations include translation, rotation, reflection, and dilation. Dilation, as in the given problem, enlarges or reduces a figure based on a center point and scale factor.For \( \triangle ABC \), the dilation transformed it using a scale factor \( k = 3 \), which was deduced from the requirement that the perimeter triples. To achieve dilation:

• Multiply each coordinate of the vertex matrix by the scale factor.
• Maintain the shape's similarity by proportionally changing each side.

This technique scales the entire shape, yielding new vertices like \( A'(0,6), B'(4.5,-4.5), C'(-7.5,0) \), and maintains the triangle's general form, only modified in size.

Perimeter Enlargement

When a problem mentions the enlargement of a shape's perimeter, it speaks to a specific transformation that increases every side by the same proportion. In dilation, if the perimeter of a shape becomes three times its original, this indicates a uniform dilation factor across all vertices.Applying this to \( \triangle ABC \), the dilation factor \( k \) reflects this uniform increase, calculated so that every coordinate is multiplied by 3. This results in a new, larger triangle with each side length and the perimeter tripled.

• Original Side Lengths and Perimeter: Calculated using distance formula.
• Dilated Side Lengths: Each side three times longer due to the factor.
• Dilated Perimeter: Clearly threefold of the original perimeter showcases a proportional enlargement.

Such calculations not only affect the size but can also influence shape analysis when considering scaled mapping and transformations.