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If coordinates of two adjacent vertices of a Paralleogram are (3,2) (1,0) and diagonals bisect each other at (2,-5) find coordinates of the other two vertices?
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Here’s a detailed guide to finding the coordinates of the other two vertices of the parallelogram.
Given Information
- Two adjacent vertices: A(3, 2) and B(1, 0)
- Diagonal intersection point (midpoint): M(2, -5)
Understanding Parallelogram Properties
- In a parallelogram, the diagonals bisect each other.
- The midpoint M of the diagonals is also the midpoint of the segment connecting the two pairs of opposite vertices.
Finding Coordinates of the Other Vertices
1. Calculate the midpoint of AB:
- Midpoint formula: M = ((x1 + x2)/2, (y1 + y2)/2)
- For A(3, 2) and B(1, 0):
- M_AB = ((3 + 1)/2, (2 + 0)/2) = (2, 1)
2. Setting up the equation for diagonals:
- Since M is the midpoint of both diagonals, we have:
- M = (P + Q)/2, where P and Q are the other two vertices.
3. Using the midpoint of AB and M:
- Since M = (2, -5), we can set up the equations for P and Q:
- (P_x + Q_x)/2 = 2
- (P_y + Q_y)/2 = -5
4. Expressing P and Q coordinates:
- Let P = (x1, y1) and Q = (x2, y2).
- From the midpoint equations:
- x1 + x2 = 4
- y1 + y2 = -10
5. Finding P and Q:
- By substituting the values of the midpoint M_AB and setting up the system, we can derive:
- P = (3 + k, 2 - m)
- Q = (1 - k, 0 + m)
- Solve for k and m using the midpoint equations to find the exact coordinates of P and Q.
Final Coordinates
- The other two vertices of the parallelogram are:
- P(5, -8)
- Q(-1, -10)
This method effectively helps in determining the vertices by leveraging the properties of a parallelogram.
Given Information
- Two adjacent vertices: A(3, 2) and B(1, 0)
- Diagonal intersection point (midpoint): M(2, -5)
Understanding Parallelogram Properties
- In a parallelogram, the diagonals bisect each other.
- The midpoint M of the diagonals is also the midpoint of the segment connecting the two pairs of opposite vertices.
Finding Coordinates of the Other Vertices
1. Calculate the midpoint of AB:
- Midpoint formula: M = ((x1 + x2)/2, (y1 + y2)/2)
- For A(3, 2) and B(1, 0):
- M_AB = ((3 + 1)/2, (2 + 0)/2) = (2, 1)
2. Setting up the equation for diagonals:
- Since M is the midpoint of both diagonals, we have:
- M = (P + Q)/2, where P and Q are the other two vertices.
3. Using the midpoint of AB and M:
- Since M = (2, -5), we can set up the equations for P and Q:
- (P_x + Q_x)/2 = 2
- (P_y + Q_y)/2 = -5
4. Expressing P and Q coordinates:
- Let P = (x1, y1) and Q = (x2, y2).
- From the midpoint equations:
- x1 + x2 = 4
- y1 + y2 = -10
5. Finding P and Q:
- By substituting the values of the midpoint M_AB and setting up the system, we can derive:
- P = (3 + k, 2 - m)
- Q = (1 - k, 0 + m)
- Solve for k and m using the midpoint equations to find the exact coordinates of P and Q.
Final Coordinates
- The other two vertices of the parallelogram are:
- P(5, -8)
- Q(-1, -10)
This method effectively helps in determining the vertices by leveraging the properties of a parallelogram.
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If coordinates of two adjacent vertices of a Paralleogram are (3,2) (1,0) and diagonals bisect each other at (2,-5) find coordinates of the other two vertices?
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If coordinates of two adjacent vertices of a Paralleogram are (3,2) (1,0) and diagonals bisect each other at (2,-5) find coordinates of the other two vertices? 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 If coordinates of two adjacent vertices of a Paralleogram are (3,2) (1,0) and diagonals bisect each other at (2,-5) find coordinates of the other two vertices? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If coordinates of two adjacent vertices of a Paralleogram are (3,2) (1,0) and diagonals bisect each other at (2,-5) find coordinates of the other two vertices?.
If coordinates of two adjacent vertices of a Paralleogram are (3,2) (1,0) and diagonals bisect each other at (2,-5) find coordinates of the other two vertices? 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 If coordinates of two adjacent vertices of a Paralleogram are (3,2) (1,0) and diagonals bisect each other at (2,-5) find coordinates of the other two vertices? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If coordinates of two adjacent vertices of a Paralleogram are (3,2) (1,0) and diagonals bisect each other at (2,-5) find coordinates of the other two vertices?.
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