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Chapter 13: Problem 26

Find the reflection of the point \((-9,7)\) over the reference line \(y=x\)

Short Answer

Expert verified
The reflection of the point \((-9,7)\) over the reference line \(y=x\) is \((7,-9)\).

Step by step solution

01

Understand the reference line

The given reference line is \(y=x\). This is a line that passes through the origin and makes an angle of 45 degrees with the x-axis and y-axis.
02

Identify the given point

The point given in the problem is \((-9,7)\). This is the point for which we need to find the reflection over the reference line.
03

Apply the formula for reflection

Apply the formula for reflection over the line \(y=x\). This formula is \((a,b) \Rightarrow (b,a)\) . So, swap the x and y coordinates. The reflection of \((-9,7)\) over the line \(y=x\) is \((7,-9)\).

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Key Concepts

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

Coordinate Geometry
In the study of geometry, coordinate geometry (or analytic geometry) is a crucial part that brings algebra and geometry together.
It allows us to understand geometry through a coordinate plane, where each point is defined by a pair of numerical coordinates.
These coordinates are often denoted as \((x, y)\) in a two-dimensional plane, such as the Cartesian coordinate system.
  • Coordinate Plane: A two-dimensional space defined by two axes, the x-axis (horizontal) and the y-axis (vertical), which intersect at the origin \((0,0)\).
  • Points: Every point on this plane has a pair of numbers which represent its distance from the origin along the x-axis and y-axis.
  • Ordered Pairs: Coordinates are presented in an order, \((x, y)\), to specify the precise location of the point on the plane.
In coordinate geometry, transformations like reflections involve changing the position of points within the plane.
This provides a clear and visual way to analyze and solve geometric problems using algebra.
Transformation
Transformation in geometry refers to the movement of a figure in the plane, which can change its position, orientation, or size.
There are several types of transformations, including translations, rotations, reflections, and dilations.
The concept of transformation is essential because it helps understand the behavior and properties of geometric figures under various conditions.
  • Types of Transformation:
    • Translation: Moves the shape without rotating or flipping it.
    • Rotation: Spins the figure around a fixed point.
    • Reflection: Flips the figure over a line, creating a mirror image.
    • Dilation: Expands or contracts the figure by multiplying the distances from a fixed point by a scale factor.
In our exercise, we focus on reflection, which specifically involves flipping a point over a line.
This transformation alters the position of the point while maintaining its shape and size.
Line of Reflection
In reflection transformations, the line of reflection acts as a virtual mirror.
It is the axis over which every point of a figure is transformed to a new location, resulting in an image that is a mirror image of the original figure.
Understanding this concept is vital to solving reflection problems accurately.
  • Mirror Line: The line equidistant between each point and its image in the reflection.
  • Perpendicular Distance: Each original point and its reflected image are equidistant from the reflection line.
  • Reflection Formula: For reflection over the line \(y = x\), the coordinates of any point \((a, b)\) transform to \((b, a)\).
In our specific problem, the line of reflection is \(y = x\), meaning each \((x, y)\) coordinate swaps places to become \((y, x)\).
This simple swap reflects the point across the designated line, maintaining symmetry.

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Most popular questions from this chapter

Determine the intersection of the solution sets of the two inequalities \(y>2\) and \(x+2 y<6\) by graphing.

Find the distance between the parallel lines corresponding to \(y=2 x+3\) and \(y=2 x+7 .\) (Hint: Start by choosing a convenient point on one of the lines.)

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