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Prove that the diagonals of a rectangle are equal and bisect each other using COORDINATE GEOMETRY.?
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Proof that the Diagonals of a Rectangle are Equal and Bisect Each Other using Coordinate Geometry
Given:
- Let the coordinates of the vertices of the rectangle be A(x1, y1), B(x2, y2), C(x3, y3), and D(x4, y4).
- Assume the rectangle is placed in such a way that sides AB and CD are parallel to the x-axis and sides AD and BC are parallel to the y-axis.
Proof:
- Step 1: Calculate the Midpoints of the Diagonals
- Step 2: Show that the Midpoints are Equal
- Step 3: Calculate the Length of the Diagonals
- Step 4: Show that the Lengths of the Diagonals are Equal
- The midpoint of diagonal AC is M(xm, ym) where xm = (x1 + x3) / 2 and ym = (y1 + y3) / 2.
- The midpoint of diagonal BD is N(xn, yn) where xn = (x2 + x4) / 2 and yn = (y2 + y4) / 2.
- To prove that the diagonals bisect each other, we need to show that M is equal to N.
- Since a rectangle is symmetric about its diagonals, the midpoints of the diagonals are equal.
- The length of diagonal AC is given by sqrt[(x3 - x1)^2 + (y3 - y1)^2].
- The length of diagonal BD is given by sqrt[(x4 - x2)^2 + (y4 - y2)^2].
- Calculate the lengths of the diagonals AC and BD using the coordinates of the vertices.
- If the lengths of the diagonals are equal, then the diagonals are equal.
Conclusion:
- Therefore, we have proved that the diagonals of a rectangle are equal and bisect each other using coordinate geometry.
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Prove that the diagonals of a rectangle are equal and bisect each other using COORDINATE GEOMETRY.?
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Prove that the diagonals of a rectangle are equal and bisect each other using COORDINATE GEOMETRY.? for Class 10 2024 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about Prove that the diagonals of a rectangle are equal and bisect each other using COORDINATE GEOMETRY.? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Prove that the diagonals of a rectangle are equal and bisect each other using COORDINATE GEOMETRY.?.
Prove that the diagonals of a rectangle are equal and bisect each other using COORDINATE GEOMETRY.? for Class 10 2024 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about Prove that the diagonals of a rectangle are equal and bisect each other using COORDINATE GEOMETRY.? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Prove that the diagonals of a rectangle are equal and bisect each other using COORDINATE GEOMETRY.?.
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