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Magazine Chapter 1: Problem 59
The \(x\) -axis is the perpendicular bisector of the line segment through \(A(2,5)\) and \(B(x, y) .\) Find \(x\) and \(y\).
Short Answer
Expert verified
Point B is at (2, -5).
Step by step solution
01
Understanding the Concept
The question states that the x-axis is the perpendicular bisector of the line segment passing through points A and B. This means that the midpoint of the line segment AB lies on the x-axis, indicating that the y-coordinate of the midpoint must be zero.
02
Finding the Midpoint of the Line Segment
The formula for the midpoint, M, of a line segment connecting points \(A(x_1, y_1)\) and \(B(x_2, y_2)\) is: \[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right). \]Here, \(A = (2,5)\) and \(B = (x,y)\). Thus, the midpoint is: \[ M = \left( \frac{2 + x}{2}, \frac{5 + y}{2} \right). \]
03
Setting the y-coordinate of the Midpoint to Zero
Since the x-axis is the perpendicular bisector, the y-coordinate of the midpoint must be zero. Therefore, \[ \frac{5 + y}{2} = 0. \]
04
Solving for y
To solve \( \frac{5 + y}{2} = 0 \), multiply both sides by 2 to eliminate the fraction: \[ 5 + y = 0. \]Subtract 5 from both sides to solve for y: \[ y = -5. \]
05
Determining the x-coordinate
Once we have found \(y = -5\), substitute back to find \(x\). Since the midpoint lies on the x-axis, it implies symmetry, leading to no additional equation directly concerning \(x\). Calculate based on symmetry alone as \(x = 2\).
06
Conclusion
The coordinates of point B are \((x, y) = (2, -5)\), maintaining the symmetry about the x-axis, with their midpoint aligned directly on it.
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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 cuts another line segment exactly in half and does so at a right angle (90 degrees) to it. In the context of coordinate geometry, such as the problem we are examining, understanding the role of a perpendicular bisector is vital. The x-axis is acting as the perpendicular bisector for the line segment connecting points A and B. This special relationship means:
- The midpoint of the segment lies on the x-axis.
- The segment splits into two equal halves.
- The x-axis is at a right angle to the segment at the midpoint.
Midpoint Formula
The midpoint formula is a powerful tool in coordinate geometry. It allows us to find the point that is exactly halfway between two given points. The formula is:\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right). \]This formula requires simple averaging of the corresponding coordinates of the two points A and B. For example:
- Point A has coordinates \((2, 5)\)
- Point B has coordinates \((x, y)\)
Symmetry in Geometry
Symmetry in geometry refers to a balanced and proportional similarity found in shapes or configurations. In coordinate geometry, this often involves reflections over axes. For the exercise at hand, the x-axis not only acts as a perpendicular bisector but also creates reflective symmetry for the line segment AB. This symmetry implies:
- The x-coordinates on either side of the midpoint are equidistant from the x-axis.
- The y-coordinate of the midpoint is zero since it's exactly mirrored.
- The problem's symmetry simplifies calculations, as balanced measurements lead us to \(x =2\) without further solving.
