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F2 circles intersect at 2 points prove that their centres lie on the perpendicular bisector of the common chord?
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F2 circles intersect at 2 points prove that their centres lie on the p...
If two circles intersect at two points, then the line that connects those two points is called the common chord of the circles. To prove that the centers of the circles lie on the perpendicular bisector of the common chord, we can use the following steps:
Draw the two circles and the common chord connecting the two points of intersection.
Draw the perpendicular bisector of the common chord. This is a line that passes through the midpoint of the common chord and is perpendicular to it.
Since the centers of the circles are equidistant from the points of intersection, they must be equidistant from the ends of the common chord. Therefore, the centers of the circles must lie on the perpendicular bisector of the common chord.
This can be proven using the Pythagorean Theorem, which states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. If we consider the centers of the two circles and one of the points of intersection as the vertices of a right triangle, then the distance between the centers is the hypotenuse, and the distance from the center of one of the circles to the common chord is one of the other two sides. Applying the Pythagorean Theorem to this right triangle, we can show that the distance from the center of one of the circles to the common chord is equal to the distance from the center of the other circle to the common chord, which means that the centers of the circles must be equidistant from the ends of the common chord. Therefore, they must lie on the perpendicular bisector of the common chord.
This proof can be visualized using the diagram below:
[Diagram showing two circles intersecting at two points, with a common chord connecting the points of intersection and a perpendicular bisector passing through the midpoint of the common chord. The centers of the circles are shown as points lying on the perpendicular bisector.]
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F2 circles intersect at 2 points prove that their centres lie on the perpendicular bisector of the common chord?
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F2 circles intersect at 2 points prove that their centres lie on the perpendicular bisector of the common chord? for Class 9 2024 is part of Class 9 preparation. The Question and answers have been prepared according to the Class 9 exam syllabus. Information about F2 circles intersect at 2 points prove that their centres lie on the perpendicular bisector of the common chord? covers all topics & solutions for Class 9 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for F2 circles intersect at 2 points prove that their centres lie on the perpendicular bisector of the common chord?.
F2 circles intersect at 2 points prove that their centres lie on the perpendicular bisector of the common chord? for Class 9 2024 is part of Class 9 preparation. The Question and answers have been prepared according to the Class 9 exam syllabus. Information about F2 circles intersect at 2 points prove that their centres lie on the perpendicular bisector of the common chord? covers all topics & solutions for Class 9 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for F2 circles intersect at 2 points prove that their centres lie on the perpendicular bisector of the common chord?.
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