Proving Circle Intersections: Centers on Perpendicular Bisector

Table of Contents:

1. Introduction
2. Explanation of the Question
3. Understanding the Common Chord
4. Proof of the Centers Being on the Perpendicular Bisector
   1. Constructing the Diagram
   2. Proving Congruence of Triangles
   3. Using Cpct to Prove Equality of Angles and Sides
   4. Proving Perpendicularity
   5. Final Conclusion and Proof of the Question
5. Conclusion

Article:

Proving the Centers of Intersecting Circles Lie on the Perpendicular Bisector of the Common Chord

In this article, we will explore question 3 of exercise 12.3 from the chapter on circles. The question states that if two circles intersect at two points, then their centers must lie on the perpendicular bisector of the common chord. We will break down the question and its concepts step by step to understand and prove this property of intersecting circles.

Introduction

When two circles intersect at two distinct points, it creates a unique relationship between their centers and the common chord. This question asks us to prove that the centers of these intersecting circles will always lie on the perpendicular bisector of the common chord. To prove this, we need to understand the concept of the common chord and how it relates to the centers of the circles.

Explanation of the Question

To begin, let us define the Scenario described in the question. We have two circles, and they intersect at two different points. This means that there are two distinct points where the circles meet. Our task is to prove that the centers of these circles lie on the perpendicular bisector of the common chord.

Understanding the Common Chord

The common chord is a line segment that joins two points of a circle. In the given question, we have two circles, and the common chord is the line segment that is common to both circles. To Visualize this, let's draw the two circles and the common chord.

Proof of the Centers Being on the Perpendicular Bisector

To prove that the centers of the circles lie on the perpendicular bisector of the common chord, we need to construct a diagram and demonstrate congruence between certain triangles.

Constructing the Diagram

In order to begin the proof, we construct a diagram with two intersecting circles. Let the centers of the circles be A and B. We draw the perpendicular bisector of the common chord passing through the centers.

Proving Congruence of Triangles

Next, we consider two triangles: triangle APB and triangle AQB. By the given information, we know that AP is equal to AQ (as they are radii of the circles), and PB is equal to QB. Therefore, we can Apply the side-side-side congruence rule to prove that these two triangles are congruent.

Using Cpct to Prove Equality of Angles and Sides

Using the congruence of the triangles, we can conclude that angle 1 is equal to angle 2. By the corresponding parts of congruent triangles (CPCD), we can also state that angle AOP is equal to angle QOA.

Proving Perpendicularity

Consider triangles AOP and AOQ. These triangles share several congruent sides and angles. Therefore, we can apply the side-angle-side congruency rule to prove that they are congruent. By Cpct, we can conclude that OP is equal to OQ.

To prove that the centers lie on the perpendicular bisector, we now consider the linear pair formed by the line segment OP and angle AOP. We know that the sum of adjacent angles forming a linear pair is equal to 180 degrees. By substituting the congruent angle AOP in place of angle QOA, we can deduce that angle AOP is equal to 90 degrees. Since angle AOP is equal to angle QOA and both sides are congruent, we can further conclude that angle QOA is also 90 degrees.

Final Conclusion and Proof of the Question

By proving that the angles formed are 90 degrees and that they bisect the common chord, we have successfully demonstrated that the centers of two intersecting circles lie on the perpendicular bisector of the common chord.

Conclusion

In this article, we have addressed question 3 from exercise 12.3 in the circle chapter. By understanding the concept of the common chord and utilizing congruent triangles, we have proven that the centers of intersecting circles will always lie on the perpendicular bisector of the common chord. This property helps us understand the relationship between the centers and the points of intersection of two circles.