Problem 4 - 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. \(A(-3,-3)\) and \(B(-3,-1)\)

Step-by-Step Solution

Answer

A'(-3, 3) and B'(-3, 1) across x-axis; A'(3, -3) and B'(3, -1) across y-axis; A'(3, 3) and B'(3, 1) across origin.

Step 1: Understanding Reflections

To reflect a point across an axis or the origin, we need to know how each point changes. Across the x-axis, the y-coordinates change sign. Across the y-axis, the x-coordinates change sign. Across the origin, both coordinates change signs.

Step 2: Reflect Across the x-axis

For point A (-3, -3), reflecting across the x-axis changes the coordinates to (-3, +3). For point B (-3, -1), reflecting across the x-axis changes the coordinates to (-3, +1). Thus, the reflected endpoints are A' (-3, 3) and B' (-3, 1).

Step 3: Reflect Across the y-axis

For point A (-3, -3), reflecting across the y-axis changes the coordinates to (+3, -3). For point B (-3, -1), reflecting across the y-axis changes the coordinates to (+3, -1). Thus, the reflected endpoints are A' (3, -3) and B' (3, -1).

Step 4: Reflect Across the Origin

For point A (-3, -3), reflecting across the origin changes the coordinates to (+3, +3). For point B (-3, -1), reflecting across the origin changes the coordinates to (+3, +1). Thus, the reflected endpoints are A' (3, 3) and B' (3, 1).

Key Concepts

Coordinate System

In geometry, a coordinate system is simply a way to use numbers to represent points on a plane. It helps us easily identify exact locations on a graph.
The most common type used is the Cartesian coordinate system. It consists of two perpendicular lines: the x-axis, which runs horizontally, and the y-axis, which runs vertically. These lines intersect at the origin, marked as (0, 0).
Each point in this system is described by two numbers, or coordinates:

• The first number is the x-coordinate, which tells us how far left or right a point is from the origin.
• The second number is the y-coordinate, indicating how far up or down a point is.

Together, the x and y coordinates give us a complete picture of where a point is located in this two-dimensional space. For example, in our exercise, the point A is at (-3, -3), which means it is 3 units to the left and 3 units down from the origin. Coordinate systems are essential in graphically representing geometric figures, like the line segment \(\overline{A B}\).

Geometric Transformations

Geometric transformations involve changing the position or size of a shape. These changes can include translations (sliding), rotations (turning), scaling (resizing), and reflections (flipping).
Specifically, reflections are a type of transformation where each point of a shape flips across a line, known as the line of reflection. This creates a mirror image of the original shape.
In our exercise, we perform reflections of a line segment by flipping its endpoints across different lines:

• Across the x-axis, where the mirror image is directly above or below.
• Across the y-axis, where the image is directly to the left or right.
• Across the origin, creating a direct opposite image in both coordinate directions.

Geometric transformations, including reflections, are crucial in understanding symmetry and other properties of geometric shapes. They also aid in visualizing and solving complex mathematical problems.

Reflection Rules

Reflection rules help us understand how to mathematically carry out reflections on points. When reflecting points, depending on the line of reflection, certain coordinates change while others remain the same.

• For reflection across the x-axis, the y-coordinate changes its sign, but the x-coordinate remains the same. So point \(A(-3, -3)\) reflects to \(A'(-3, +3)\).
• For reflection across the y-axis, the x-coordinate changes its sign, leaving the y-coordinate untouched. Thus, \(B(-3, -1)\) becomes \(B'(3, -1)\).
• Reflecting across the origin flips both coordinates, like \(A(-3, -3)\) becoming \(A'(3, 3)\).

During reflections, the distance from each point to the line of reflection is the same on both sides, maintaining equal spacing. These rules are indispensable in accurately finding the new coordinates of reflected points. Understanding these basics gives us a foundation not only in geometry but in numerous applications in physics, engineering, and computer graphics.