Problem 5 - Precalculus

Question

The endpoints of a line segment \(\overline{A B}\) are given. Sketch the reflection of \(\overline{A B}\) about (a) the \(x\) -axis; (b) the \(y\) -axis; and (c) the origin. \(A(0,1)\) and \(B(3,1)\)

Step-by-Step Solution

Answer

Reflections: (a) A'(0,-1), B'(3,-1); (b) A'(0,1), B'(-3,1); (c) A'(0,-1), B'(-3,-1).

Step 1: Understand the Problem

We have line segment \(\overline{AB}\) with endpoints \(A(0,1)\) and \(B(3,1)\). We need to reflect this line segment in three different ways: about the \(x\)-axis, the \(y\)-axis, and the origin.

Step 2: Reflection across the x-axis

When reflecting a point across the \(x\)-axis, the \(x\)-coordinate remains the same, while the \(y\)-coordinate changes sign. Thus, the reflection of point \(A(0,1)\) is \(A'(0,-1)\), and the reflection of point \(B(3,1)\) is \(B'(3,-1)\).

Step 3: Reflection across the y-axis

Reflecting across the \(y\)-axis involves changing the sign of the \(x\)-coordinate, while the \(y\)-coordinate remains the same. Hence, the reflection of point \(A(0,1)\) is \(A'(0,1)\) (since it's on the y-axis and doesn't move), and the reflection of point \(B(3,1)\) is \(B'(-3,1)\).

Step 4: Reflection across the origin

For a reflection across the origin, both the \(x\) and \(y\) coordinates change sign. Therefore, the reflection of \(A(0,1)\) is \(A'(0,-1)\), and the reflection of \(B(3,1)\) is \(B'(-3,-1)\).

Step 5: Sketch the Reflections

Using the calculated points, sketch the original line segment \(\overline{AB}\) and its reflections. For each case (a), (b), and (c), plot points \(A'\) and \(B'\) and connect them to show the reflected line segment.

Key Concepts

Line Reflection

Line reflection is a fascinating transformation in geometry. It's like creating a mirror image of a shape or line over a specific axis. The concept revolves around flipping a point or figure to the opposite side of a given line. This is an important aspect as it maintains the shape's size and form while changing its position.

• When a line segment is reflected, each point on the segment flips over the line of reflection.
• The reflected line segment has points directly opposite and equidistant from the line of reflection.

Line reflections are crucial in understanding symmetry and spatial reasoning. They show how objects can transform without altering their fundamental properties.

Coordinates Transformation

Coordinates transformation involves changing the position of a point or shape on the coordinate plane. It allows us to manipulate figures through various transformations, such as translations, rotations, and reflections.

• For reflection transformations, coordinates change according to which axis they are reflected across.
• These transformations enable us to move or flip shapes and still retain their initial dimensions and structure.

Understanding coordinates transformation is essential for graphing and analyzing geometric figures. It helps students see how positions and orientations of shapes can be altered in a plane.

Reflection Across Axes

When a figure is reflected across axes, only certain coordinates change while others remain the same.Reflection across the x-axis is when:

• The x-coordinate remains unchanged.
• The y-coordinate changes its sign, turning a point located at \((x, y)\) into \((x, -y)\).

Reflection across the y-axis results in:

• The y-coordinate remains unchanged, while
• The x-coordinate changes its sign. This transforms a point originally at \((x, y)\) to \((-x, y)\).

Reflecting across axes is a simple yet powerful tool for creating symmetrical designs and analyzing geometrical problems.

Origin Reflection

Origin reflection is when you change both the x and y coordinates' signs. This is akin to performing two reflections—over the x-axis and the y-axis—sequentially.

• Origin reflection turns a point located at \((x, y)\) into \((-x, -y)\).
• It flips the figure entirely across the origin, creating a direct opposite in terms of position on the coordinate plane.

Using origin reflection makes it possible to explore geometric relationships by shifting perspectives across the entire plane. This transformation shows how a shape can be completely inverted without distortion, revealing more about the interplay between geometry's core elements.