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

Step-by-Step Solution

Answer

(a) Across x-axis: A'(1,-4), B'(3,-1); (b) Across y-axis: A'(-1,4), B'(-3,1); (c) Across origin: A'(-1,-4), B'(-3,-1).

Step 1: Identify Initial Coordinates

The endpoints of the line segment are given as \(A(1,4)\) and \(B(3,1)\). We'll use these coordinates for reflecting the line segment about different axes and the origin.

Step 2: Reflection Across the x-axis

To reflect a point \((x, y)\) across the \(x\)-axis, we change its \(y\)-coordinate to \(-y\). Thus, the reflected coordinates of \(A(1,4)\) become \(A'(1,-4)\), and \(B(3,1)\) become \(B'(3,-1)\). The reflected line segment across the \(x\)-axis has endpoints \(A'(1,-4)\) and \(B'(3,-1)\).

Step 3: Reflection Across the y-axis

To reflect a point \((x, y)\) across the \(y\)-axis, we change its \(x\)-coordinate to \(-x\). Thus, the reflected coordinates of \(A(1,4)\) become \(A'(-1,4)\), and \(B(3,1)\) become \(B'(-3,1)\). The reflected line segment across the \(y\)-axis has endpoints \(A'(-1,4)\) and \(B'(-3,1)\).

Step 4: Reflection Across the Origin

To reflect a point \((x, y)\) across the origin, we change both coordinates to their opposites, \((-x, -y)\). Thus, the reflected coordinates of \(A(1,4)\) become \(A'(-1,-4)\), and \(B(3,1)\) become \(B'(-3,-1)\). The reflected line segment across the origin has endpoints \(A'(-1,-4)\) and \(B'(-3,-1)\).

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study of geometric figures through a coordinate system. This branch of geometry links algebra and geometry, enabling us to solve geometric problems in an algebraic way. In coordinate geometry, any point can be represented as an ordered pair

• The first number represents the horizontal position (x-coordinate).
• The second number represents the vertical position (y-coordinate).

By assigning coordinates to geometric figures, such as line segments, we can easily calculate distances, midpoints, slopes, and perform various transformations. This systematic approach helps us visualize and solve problems more effectively.

Reflection Across Axes

Reflection is a type of transformation that flips a figure over a line, in this case, the axes.

• When a point \(x,y\) is reflected across the x-axis, its y-coordinate changes sign, resulting in the point \(x,-y\).
• For reflection across the y-axis, the x-coordinate changes sign, yielding the point \(-x,y\).

These flips can be easily visualized on a coordinate grid, where the original point and its reflection are equidistant from the axis of reflection. Reflection is useful for solving problems that require symmetry or mirroring of shapes.

Origin Symmetry

Origin symmetry means that a point or shape is mirrored across both the x and y axes simultaneously. In mathematical terms, if a point \(x,y\) is reflected through the origin, its coordinates become \(-x,-y\). Origin symmetry is characteristic of functions or shapes that, when rotated 180 degrees about the origin, look identical to the original.

• This transformation involves a direct flip across both axes.
• It's often used in problems dealing with complex symmetry.

Understanding origin symmetry is important for analyzing the behaviors of geometric shapes and functions in coordinate geometry.

Line Segment

A line segment in geometry is a part of a line bounded by two distinct endpoints. It is the shortest path connecting these two points on a plane. In coordinate geometry, a line segment is described as the set of all points \( (x, y) \) that lie between and include the endpoints. For a line segment \( \overline{AB} \) with endpoints \( A(x_1, y_1) \) and \( B(x_2, y_2) \), its length can be calculated using the distance formula: \[\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]Line segments become particularly interesting when they undergo transformations such as reflection, as they maintain properties such as length, even as their orientation changes.