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What is the Equation of the Perpendicular Bisector of a Line Segment?
Grade Level:
Class 10
AI/ML, Physics, Biotechnology, Space Technology, Chemistry, Engineering, Medicine
Definition
What is it?
The equation of the perpendicular bisector of a line segment is a mathematical rule that describes a line. This special line cuts another line segment exactly in half at its midpoint, and it also forms a perfect 90-degree angle with that segment.
Simple Example
Quick Example
Imagine you have a straight road connecting two villages, say Village A and Village B. The perpendicular bisector of this road would be a new path. This new path would start exactly in the middle of the road and cross it perfectly straight, like a 'T' junction, making sure both sides are equal. So, any point on this new path is equally far from Village A and Village B.
Worked Example
Step-by-Step
Let's find the equation of the perpendicular bisector of the line segment joining points A(2, 3) and B(6, 7).
Step 1: Find the midpoint of the line segment AB.
Midpoint M = ((x1 + x2)/2, (y1 + y2)/2)
M = ((2 + 6)/2, (3 + 7)/2)
M = (8/2, 10/2)
M = (4, 5)
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Step 2: Find the slope of the line segment AB.
Slope m_AB = (y2 - y1) / (x2 - x1)
m_AB = (7 - 3) / (6 - 2)
m_AB = 4 / 4
m_AB = 1
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Step 3: Find the slope of the perpendicular bisector.
The product of slopes of two perpendicular lines is -1. So, m_perpendicular = -1 / m_AB.
m_perpendicular = -1 / 1
m_perpendicular = -1
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Step 4: Use the point-slope form (y - y1 = m(x - x1)) to find the equation of the perpendicular bisector. We use the midpoint (4, 5) and the perpendicular slope (-1).
y - 5 = -1(x - 4)
y - 5 = -x + 4
y = -x + 4 + 5
y = -x + 9
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Answer: The equation of the perpendicular bisector is y = -x + 9.
Why It Matters
Understanding perpendicular bisectors is crucial in fields like engineering and robotics to design stable structures and robot movements. It helps in creating balanced designs in architecture and even in computer graphics for smooth animations. This concept can lead to careers in designing smart city layouts or even developing AI for autonomous vehicles.
Common Mistakes
MISTAKE: Students sometimes forget to find the midpoint and instead use one of the endpoints for the equation. | CORRECTION: Always find the midpoint first, as the perpendicular bisector MUST pass through the middle of the segment.
MISTAKE: Incorrectly calculating the perpendicular slope by just taking the negative of the original slope, not the negative reciprocal. | CORRECTION: Remember, if the original slope is 'm', the perpendicular slope is '-1/m'. Don't forget to flip the fraction!
MISTAKE: Making calculation errors when finding the midpoint coordinates or when substituting values into the slope formula. | CORRECTION: Double-check your addition, subtraction, and division steps carefully, especially with negative numbers.
Practice Questions
Try It Yourself
QUESTION: Find the equation of the perpendicular bisector of the line segment joining P(1, 2) and Q(3, 4). | ANSWER: y = -x + 5
QUESTION: A line segment connects points C(-2, 5) and D(4, -1). Determine the equation of its perpendicular bisector. | ANSWER: y = x + 1
QUESTION: The vertices of a triangle are A(0,0), B(6,0), and C(3,8). Find the equation of the perpendicular bisector of side AB. | ANSWER: x = 3
MCQ
Quick Quiz
Which of the following conditions must a line satisfy to be the perpendicular bisector of a line segment?
It must be parallel to the line segment.
It must pass through one endpoint of the line segment.
It must pass through the midpoint and be perpendicular to the line segment.
It must have the same slope as the line segment.
The Correct Answer Is:
C
A perpendicular bisector, by definition, cuts the line segment exactly in half (passes through the midpoint) and forms a 90-degree angle with it (is perpendicular). Options A, B, and D describe incorrect properties.
Real World Connection
In the Real World
In city planning, if you want to build a new hospital that is equally accessible from two major bus stops, you'd find the perpendicular bisector of the line connecting those stops. Any location on this bisector would be the same distance from both stops. This helps in fair resource distribution and efficient service delivery, much like how food delivery apps like Zomato or Swiggy might optimize delivery zones.
Key Vocabulary
Key Terms
MIDPOINT: The exact middle point of a line segment. | SLOPE: A measure of the steepness or gradient of a line. | PERPENDICULAR: Forming a 90-degree angle. | RECIPROCAL: The result of dividing 1 by a number (e.g., reciprocal of 2 is 1/2). | EQUATION: A mathematical statement showing two expressions are equal.
What's Next
What to Learn Next
Great job understanding perpendicular bisectors! Next, you can explore 'Circumcenter of a Triangle'. This builds on what you learned because the circumcenter is found by intersecting the perpendicular bisectors of a triangle's sides, which is super interesting!
