Problem 44 On a coordinate plane, draw tria... [FREE SOLUTION]

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Chapter 2: Problem 44

On a coordinate plane, draw triangle \(A B C\) with vertices at \(A(3,1), B(4,2),\) and \(C(2,4) .\) Then graph and describe each new triangle formed in Exercises \(41-44\). Subtract 4 from each coordinate of the vertices in triangle \(A B C\).

Short Answer

Expert verified

The new triangle vertices are \(A'(-1, -3)\), \(B'(0, -2)\), \(C'(-2, 0)\).

Step by step solution

Identify the Original Coordinates

The vertices of triangle \(ABC\) are given as \(A(3,1)\), \(B(4,2)\), and \(C(2,4)\).

Subtract 4 from Each Coordinate

For each vertex of the triangle, we subtract 4 from both the x-coordinate and the y-coordinate.

Calculate New Coordinates for Vertex A

The original coordinate for \(A\) is \((3,1)\). Subtract 4 from each:\(3 - 4 = -1, \quad 1 - 4 = -3\)Thus, the new coordinate for \(A\) is \((-1, -3)\).

Calculate New Coordinates for Vertex B

The original coordinate for \(B\) is \((4,2)\). Subtract 4 from each:\(4 - 4 = 0, \quad 2 - 4 = -2\)Thus, the new coordinate for \(B\) is \((0, -2)\).

Calculate New Coordinates for Vertex C

The original coordinate for \(C\) is \((2,4)\). Subtract 4 from each:\(2 - 4 = -2, \quad 4 - 4 = 0\)Thus, the new coordinate for \(C\) is \((-2, 0)\).

Graph the Transformed Triangle

Plot the new coordinates on the coordinate plane to form the new triangle: \(A'(-1, -3)\), \(B'(0, -2)\), and \(C'(-2, 0)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Coordinate Plane

The coordinate plane is a two-dimensional surface on which we can plot points, lines, and shapes. It consists of two perpendicular number lines intersecting at a point called the origin. The horizontal number line is known as the x-axis and the vertical one is the y-axis.
Each point on this plane is identified by an ordered pair \(x, y\), which tells us the position of the point relative to the origin. The x-coordinate shows how far left or right the point is from the y-axis, while the y-coordinate indicates how far up or down the point is from the x-axis.
Some key characteristics of the coordinate plane include:

Quadrants: The plane is divided into four quadrants. Points in the first quadrant have positive x and y coordinates.
Positive and Negative Values: Above the x-axis, y-coordinates are positive, and to the right of the y-axis, x-coordinates are positive.
Graphical Representation: By plotting points and drawing lines or shapes, we can visually interpret algebraic relationships.

Graphing Transformations

Graphing transformations involve changing the position or size of shapes on the coordinate plane. These changes can be translations, reflections, rotations, or dilations. In this exercise, we focus on translations, specifically shifting the entire triangle by subtracting 4 from both x and y coordinates.
Here鈥檚 how transformations can affect a graph:

Translation: Moves every point of a shape by the same distance in a given direction. Here, subtracting 4 from each coordinate shifts the triangle left and down.
Reflection: Flips a shape over a specific line, like a mirror image.
Rotation: Turns a shape around a fixed point, changing its orientation.
Dilation: Changes the size of the shape but keeps the same center point.

Performing transformations allows us to explore properties that remain unchanged, such as the shape's orientation and angles, while altering other aspects like position.

Triangle Vertices

The vertices of a triangle are the points where its sides intersect. In coordinate geometry, these points are represented by ordered pairs. For triangle \(ABC\), the points are given as \(A(3,1)\), \(B(4,2)\), and \(C(2,4)\). These vertices define the triangle's size, shape, and position on the coordinate plane.
When graphing or transforming a triangle, understanding the importance of its vertices is crucial:

Coordinates: Each vertex is positioned by its x and y coordinates, indicating its precise location.
Transformations: Changes to the vertices鈥� coordinates will shift or reshape the triangle.
Visualization: Plotting these points allows us to draw the triangle and see its geometric properties, like side length and angle.

By modifying the vertices' coordinates, we perform transformations that affect the triangle while maintaining its fundamental structure.

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91影视

On a coordinate plane, draw triangle \(A B C\) with vertices at \(A(3,1), B(4,2),\) and \(C(2,4) .\) Then graph and describe each new triangle formed in Exercises \(41-44\). Subtract 4 from each coordinate of the vertices in triangle \(A B C\).

Short Answer

Step by step solution

Identify the Original Coordinates

Subtract 4 from Each Coordinate

Calculate New Coordinates for Vertex A

Calculate New Coordinates for Vertex B

Calculate New Coordinates for Vertex C

Graph the Transformed Triangle

Key Concepts

Coordinate Plane

Graphing Transformations

Triangle Vertices

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