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Chapter 9: Problem 29

The coordinates of the vertices of \(\triangle J K L\) are \(J(-3,4), K(0,5),\) and \(L(5,10)\). CAN'T COPY THE GRAPH Graph the image of \(\triangle J K L\) after a reflection in \(x=2\) and one in \(x=6\)

Short Answer

Expert verified
The vertices after reflections are J''(5,4), K''(8,5), L''(3,10).

Step by step solution

01

Understand the Concept of Reflection

When a point is reflected over a line, its image is equidistant from the line on the opposite side. In this problem, we are reflecting the triangle first over the line \(x=2\) and then over the line \(x=6\). The reflection process involves finding the new coordinates of each vertex after these reflections.
02

Reflect Over x=2 Line

Reflect each vertex over the x=2 line:\(-3\) to the left of 2, J's reflection is at \((-3)+(2-(-3)) \rightarrow x = 7\): J' is at (7, 4).\(0\) reflecting across 2 is found by \(2x - 0 = 2 \rightarrow K' \, will be \, (4, 5)\).Reflect L \(5\) across 2, \(2x - 5 = 2 \rightarrow x = -1\), L' is (9,10). These are the vertices of the image after the first reflection, J'(7,4), K'(4,5), L'(9,10).
03

Reflect Over x=6 Line

Reflect each vertex from Step 2 over the line \(x=6\):J' is 7 and reflecting over 6, calculate New J: 2x -7 = 6, so x' = 5K' is 4, reflect over 6: 2x-4 = 6, x'=8.Reflecting L' is 9, reflect over 6, using 2x -9 = 6 x'=3.Hence the final vertices after both reflections are J''=(5,4), K''=(8,5), L''=(3,10).
04

Plot the Image

Plot the original triangle \(J(-3,4), K(0,5), L(5,10)\) and the new triangle \(J''(5,4), K''(8,5), L''(3,10)\) on a coordinate plane. Use the line x=2 and x=6 to help visualize the reflections.

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Key Concepts

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

Reflection
In coordinate geometry, reflection is a kind of transformation that flips a shape over a line, creating a mirror image. This line, called the "line of reflection," acts as the mirror. To reflect a point over a line, you need to determine its distance from the line and place it equally far on the opposite side. In our exercise, \triangle JKL\ is reflected over the lines \(x=2\) and \(x=6\).
  • For the first reflection over \(x=2\), each vertex's x-coordinate changes based on its distance from the line. If a vertex is at \(x=a\) and the line of reflection is \(x=b\), the reflected x-coordinate is \(2b-a\).
  • The same process applies for reflection over \(x=6\) using the coordinates after the first reflection.
Reflection helps in understanding symmetry and congruence in shapes, crucial concepts in geometry.
Graphing
Graphing in coordinate geometry involves plotting points on a coordinate plane to form various shapes, like triangles. Each point has an x-coordinate and a y-coordinate. When graphing, it's crucial to place each point correctly based on these coordinates.
  • To graph a triangle, plot each of its vertices on the coordinate plane and connect them with straight lines.
  • For example, in our exercise, the original triangle \( \triangle J K L \) has vertices \((-3,4)\), \((0,5)\), and \((5,10)\).
  • After reflecting these vertices, new points are plotted to show the triangle's new position.
By visualizing these points and their connections, we can easily see the effects of transformations and how the triangle's position changes.
Vertices
Vertices are the corner points of geometric shapes. In our triangle \(\triangle J K L\), the vertices are \(J(-3,4)\), \(K(0,5)\), and \(L(5,10)\). These points define the shape and size of the triangle.
  • A vertex is a crucial reference point in transformations like reflection and rotations.
  • In our exercise, each of these vertices shifts positions after each reflection to new coordinates.
Understanding vertices is key because they help maintain the shape's structure when transformations like reflections are applied. By accurately plotting these new vertices, the transformed shape can be easily constructed.
Coordinate Plane
The coordinate plane is a two-dimensional surface for graphing points, lines, and shapes. It consists of two axes: the horizontal x-axis and the vertical y-axis. The intersection of these axes is the origin, \((0,0)\). Each point on this plane is identified by a pair of coordinates, \((x,y)\).
  • In our exercise, we plot the original and reflected positions of \(\triangle J K L \) on this plane.
  • The lines \(x=2\) and \(x=6\) are vertical on the coordinate plane and serve as mirrors for the reflections.
  • Understanding how to plot and read points on this plane is vital for visualizing geometric transformations.
By mastering the use of the coordinate plane, you can effectively solve geometry problems like reflections and gain a clearer understanding of shapes and their transformations.

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