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Chapter 3: Problem 13

Find the indicated coordinates. \(P\) is the point \((3,2) .\) Locate point \(Q\) such that the \(x\) -axis is the perpendicular bisector of the line segment joining \(P\) and \(Q\).

Short Answer

Expert verified
The coordinates of Q are (3, -2).

Step by step solution

01

Understand the Problem

We need to find point \( Q \) such that the \( x \)-axis is the perpendicular bisector of the line segment joining \( P \) and \( Q \). This means the \( x \)-coordinates of \( P \) and \( Q \) will be the same, and the \( y \)-coordinates will be opposites.
02

Determine Coordinates of \( Q \)

For the \( x \)-coordinates, since the \( x \)-axis is the perpendicular bisector, the \( x \)-coordinate of \( Q \) will be the same as that of \( P \). So \( x_Q = 3 \).For the \( y \)-coordinates, reflection over the \( x \)-axis means that \( y_Q = -y_P \). Given \( y_P = 2 \), we have \( y_Q = -2 \).
03

Conclusion: Find \( Q \)

Combining the \( x \)- and \( y \)-coordinates, we find that \( Q \) is \( (3, -2) \).

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Key Concepts

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

Perpendicular Bisector
A perpendicular bisector is a line that divides another line segment into two equal parts at a 90-degree angle. In coordinate geometry, it has a unique property where it finds the midpoint of a segment and intersects it perpendicularly. To clearly grasp this, remember that if you have a line segment joining two points, the perpendicular bisector will:
  • Cut the segment into equal halves.
  • Meet the segment at a 90-degree angle.
For this exercise, our perpendicular bisector is the x-axis. So, if you have two points on a coordinate plane, and the x-axis is the perpendicular bisector, it means each point will fall on either side of the x-axis with equal distance from it.
Reflection
Reflection in coordinate geometry involves flipping a point over a line to create a mirror image of the original point. Consider the x-axis in your coordinate plane as a mirror. When you reflect a point over the x-axis, its x-coordinate stays the same, while the y-coordinate changes its sign.Here's how you determine the reflected point:
  • If you start with a point \( (x, y) \), the reflection over the x-axis will result in a point \( (x, -y) \).
  • The distance from the original point to the axis equals the distance from the axis to its reflection.
In the given exercise, point Q was found by reflecting point P \( (3, 2) \) over the x-axis, resulting in Q \( (3, -2) \). The x-coordinate remains constant while the y-coordinate flips.
Coordinate Plane
The coordinate plane is an essential tool in coordinate geometry, composed of two perpendicular lines that intersect at a right angle. These two lines are known as the x-axis (horizontal) and the y-axis (vertical). Together, they divide the plane into four quadrants for locating points with precision.Key features of the coordinate plane:
  • Points are represented as \( (x, y) \), with x being the horizontal placement and y the vertical.
  • The center point where the axes intersect is named the origin, denoted as \( (0, 0) \).
  • Knowing which quadrants points fall into helps determine their positions relative to each axis.
In our problem, understanding the coordinate plane is crucial. With point P given as \( (3, 2) \), we find Q by reflecting P over the x-axis while using the coordinate plane鈥檚 properties effectively to guide the calculation.

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