Problem 3 - 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. \(A(-2,-3)\) and \(B(2,-1)\)

Step-by-Step Solution

Answer

(a) Reflection over x-axis: \((-2, 3)\) to \((2, 1)\). (b) y-axis: \((2, -3)\) to \((-2, -1)\). (c) Origin: \((2, 3)\) to \((-2, 1)\).

Step 1: Reflect over the x-axis

To reflect a point over the x-axis, you keep the x-coordinate the same and change the sign of the y-coordinate. For point \(A(-2, -3)\), the reflection over the x-axis is \((-2, 3)\). For point \(B(2, -1)\), the reflection over the x-axis is \((2, 1)\). Thus, the reflected line segment over the x-axis is between \((-2, 3)\) and \((2, 1)\).

Step 2: Reflect over the y-axis

To reflect a point over the y-axis, you change the sign of the x-coordinate and keep the y-coordinate the same. Therefore, for point \(A(-2, -3)\), the reflection over the y-axis is \((2, -3)\). For point \(B(2, -1)\), the reflection over the y-axis is \((-2, -1)\). The reflected segment over the y-axis is between \((2, -3)\) and \((-2, -1)\).

Step 3: Reflect over the origin

To reflect a point over the origin, change the signs of both the x and y coordinates. For \(A(-2, -3)\), reflecting over the origin gives \((2, 3)\). For \(B(2, -1)\), it becomes \((-2, 1)\). The reflected line segment over the origin is between \((2, 3)\) and \((-2, 1)\).

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, uses a coordinate system to describe geometric figures and relationships. It combines algebra and geometry, letting us work with figures precisely. Here, every point on the plane is determined by a pair of numbers, the coordinates, written as \((x, y)\). These coordinate pairs indicate the position by measuring from a reference point, the origin \((0, 0)\).

• X-coordinate: This indicates the point's horizontal position. A positive value means it is to the right of the origin, and a negative value denotes it is to the left.
• Y-coordinate: This shows the point's vertical position. A positive value lies above the origin, while a negative value is below.

Coordinate geometry makes it easy to calculate distances between points, the slope of lines, and many other geometric concepts. Additionally, transformations, such as reflection, can be determined straightforwardly using coordinate rules.

Reflection in the Coordinate Plane

Reflection in the coordinate plane involves flipping a figure over a specific line, such as the x-axis, y-axis, or the origin. This transformation requires a basic rule for changing the coordinates of the figures involved, effectively moving their positions. When reflected:- **Over the x-axis:** Keep the x-coordinate the same, and change the sign of the y-coordinate. For example, \((a, b) \) ➔ \((a, -b)\).- **Over the y-axis:** Keep the y-coordinate constant and change the sign of the x-coordinate. For example, \((a, b)\) ➔ \((-a, b)\).- **Over the origin:** Change the sign for both coordinates, \((a, b)\) ➔ \((-a, -b)\).These rules allow you to easily sketch the reflection across any of these lines by simply modifying their coordinates. Reflections result in a mirror image of the original figure in the plane.

Transformations in Geometry

Transformations in geometry are operations that alter the position or size of geometric figures. Reflections, translations, rotations, and dilations are the four main types. Each transformation involves a systematic change using rules that apply to the figure’s coordinates.

• Reflections: As seen used earlier, reflections create a mirrored position of a point or line across an axis or the origin.
• Translations: This slides the figure from one place to another by adding a set value to the coordinates.
• Rotations: Involves turning the figure around a fixed point, with its coordinates adjusted according to a specific angle.
• Dilations: Change the size of a figure by a scale factor, affecting its coordinates proportionally.

Each transformation has a unique impact on the figure but maintains certain properties, such as symmetry, angles, and proportionality of sides. Understanding these transformations equips you with powerful tools to manipulate and explore geometric figures efficiently on the coordinate plane.