Problem 44 - Algebra 2

Question

For Exercises 44 and \(45,\) use the following information. The vertices of \(\triangle A B C\) are \(A(-2,1), B(1,2)\) and \(C(2,-3) .\) The triangle is dilated so that its perimeter is 2\(\frac{1}{2}\) times the original perimeter. Write the coordinates of \(\triangle A B C\) in a vertex matrix.

Step-by-Step Solution

Answer

The vertex matrix for \(\triangle ABC\) is \( \begin{bmatrix} -2 & 1 & 2 \\ 1 & 2 & -3 \\ \end{bmatrix} \).

Step 1: Identify Vertices

The vertices given for triangle \(\triangle ABC\) are \(A(-2,1)\), \(B(1,2)\), and \(C(2,-3)\). These coordinates will be used to form the vertex matrix.

Step 2: Formulate Vertex Matrix

To create the vertex matrix, list the coordinates of the vertices as columns of a matrix. Each column represents a vertex, with the first row representing the x-coordinates and the second row the y-coordinates.\[\begin{bmatrix}-2 & 1 & 2 \1 & 2 & -3 \\end{bmatrix}\]This matrix represents the coordinates of vertices \(A\), \(B\), and \(C\) in \(\triangle ABC\).

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of geometry where we use a coordinate system to describe the relationships between points, lines, and shapes. One key aspect of coordinate geometry is locating points on the Cartesian plane, which shows the position of points through pairs of numbers, called coordinates. The coordinates are usually written as \(x, y\), where

• \(x\) is the horizontal distance from the origin (the vertical axis).
• \(y\) is the vertical distance from the origin (the horizontal axis).

In the example of triangle \(\triangle ABC\), each vertex of the triangle is expressed with coordinates \(A(-2,1)\), \(B(1,2)\), and \(C(2,-3)\). These numbers provide essential information about the exact spot each point occupies on the plane, which allows us to map, calculate distances, and comprehend the spatial relationships within the triangle.

Dilation in Geometry

Dilation is a transformation that produces an image that is the same shape as the original but is a different size. The transformation involves scaling objects by a certain factor known as the scale factor. When understanding dilation in geometry, remember:

• A scale factor greater than 1 enlarges the shape.
• A scale factor between 0 and 1 shrinks the shape.
• The angles remain the same, preserving the shape's geometry.

For triangle \(\triangle ABC\) in the exercise, it is dilated so that its perimeter becomes 2\(\frac{1}{2}\) times the original perimeter. This means that every linear measurement of the triangle, like side lengths, increases by the same ratio of 2.5. However, the ratios of the sides and angles within the triangle itself do not change due to the dilation process.

Triangular Matrices

While vertex matrices are not triangular matrices in the mathematical sense, understanding how triangular matrices work helps in broader matrix applications. A triangular matrix is a special type of square matrix where all the elements above or below the main diagonal are zero. They come in two forms:

• Upper triangular, where all elements below the main diagonal are zero.
• Lower triangular, where all elements above the main diagonal are zero.

These matrices are especially useful in solving systems of linear equations. However, in the context of the original exercise, the focus was on vertex matrices, not triangular ones. The vertex matrix creates an organized way to handle the coordinates: \[\begin{bmatrix}-2 & 1 & 2 \1 & 2 & -3 \\end{bmatrix}\]This matrix does not fit the criteria of a triangular matrix, as it's not square, but it serves to neatly display the coordinates of vertices \(A, B,\) and \(C\).