In A Parallelogram OABC, Given The Coordinates Of A (2,6) And C(3,-3), How Do You Find The Coordinates Of B?

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Introduction

In geometry, a parallelogram is a quadrilateral with two pairs of parallel sides. One of the fundamental properties of parallelograms is that their opposite sides are equal in length, and their opposite angles are equal. Additionally, the diagonals of a parallelogram bisect each other. This means that the point of intersection of the diagonals is the midpoint of each diagonal. In this article, we will delve into the problem of finding the coordinates of point B in a parallelogram OABC, given the coordinates of points A and C. This problem combines geometric principles with coordinate geometry, providing a rich exploration of how algebraic tools can be used to solve geometric problems. Understanding parallelograms and their properties is crucial not only in geometry but also in various fields such as engineering, architecture, and computer graphics. The ability to determine unknown coordinates based on known geometric properties is a valuable skill that highlights the interplay between geometry and algebra. Let's embark on a step-by-step journey to solve this problem, enhancing our understanding of parallelograms and coordinate geometry along the way.

Understanding Parallelograms and Coordinate Geometry

Before we dive into the solution, let's ensure we have a solid grasp of the underlying concepts. A parallelogram is a quadrilateral where opposite sides are parallel and equal in length. This symmetry leads to several important properties: opposite angles are equal, and the diagonals bisect each other. This bisection property is key to our solution, as it means the midpoint of diagonal AC is the same as the midpoint of diagonal OB. In coordinate geometry, we represent points in a plane using ordered pairs (x, y). These coordinates allow us to describe geometric shapes algebraically. The midpoint formula, a cornerstone of coordinate geometry, states that the midpoint of a line segment connecting points (x₁, y₁) and (x₂, y₂) is ((x₁ + x₂)/2, (y₁ + y₂)/2). This formula will be instrumental in finding the coordinates of point B. By combining the geometric properties of parallelograms with the algebraic tools of coordinate geometry, we can solve a variety of problems, including finding unknown coordinates. This approach underscores the power of mathematical reasoning and the interconnectedness of different mathematical concepts. The problem at hand provides an excellent opportunity to apply these concepts and deepen our understanding of both geometry and algebra.

Problem Statement: Parallelogram OABC

Let's clearly define the problem we aim to solve. We are given a parallelogram OABC, where O is the origin (0,0), A has coordinates (2,6), and C has coordinates (3,-3). Our task is to determine the coordinates of point B. This problem elegantly combines geometric properties with coordinate geometry. To solve it, we will leverage the property that the diagonals of a parallelogram bisect each other. This means that the midpoint of diagonal AC is the same as the midpoint of diagonal OB. By applying the midpoint formula and using the given coordinates, we can set up equations to solve for the unknown coordinates of point B. The problem's simplicity belies its depth, as it requires a solid understanding of both geometric principles and algebraic techniques. Visualizing the parallelogram in a coordinate plane can be helpful in understanding the relationships between the points. The strategic application of the midpoint formula, combined with the properties of parallelograms, will lead us to the solution. This problem serves as a valuable exercise in mathematical problem-solving, reinforcing the importance of clear problem definition and the application of relevant concepts.

Solution: Finding the Coordinates of B

To find the coordinates of point B, we will use the property that the diagonals of a parallelogram bisect each other. This means the midpoint of AC is the same as the midpoint of OB. Let the coordinates of B be (x, y). First, we find the midpoint of AC. The midpoint formula states that the midpoint of a line segment connecting points (x₁, y₁) and (x₂, y₂) is ((x₁ + x₂)/2, (y₁ + y₂)/2). Applying this to points A(2, 6) and C(3, -3), we get the midpoint of AC as: Midpoint of AC = ((2 + 3)/2, (6 + (-3))/2) = (5/2, 3/2). Next, we find the midpoint of OB. Since O is the origin (0, 0) and B is (x, y), the midpoint of OB is: Midpoint of OB = ((0 + x)/2, (0 + y)/2) = (x/2, y/2). Now, we equate the midpoints of AC and OB, since they are the same point: (x/2, y/2) = (5/2, 3/2). This gives us two equations: x/2 = 5/2 and y/2 = 3/2. Solving for x, we multiply both sides of the first equation by 2: x = 5. Solving for y, we multiply both sides of the second equation by 2: y = 3. Therefore, the coordinates of point B are (5, 3). This solution elegantly demonstrates the power of combining geometric properties with algebraic techniques to solve problems in coordinate geometry. The strategic use of the midpoint formula, along with the understanding of parallelogram properties, has allowed us to determine the unknown coordinates of point B.

Step-by-Step Calculation

Let's break down the calculation process step-by-step to ensure clarity.

  1. Identify the given information: We have parallelogram OABC with O(0,0), A(2,6), and C(3,-3). We need to find the coordinates of B.
  2. Recall the properties of a parallelogram: The diagonals of a parallelogram bisect each other. This means the midpoint of AC is the same as the midpoint of OB.
  3. Apply the midpoint formula to find the midpoint of AC: Midpoint of AC = ((x₁ + x₂)/2, (y₁ + y₂)/2) = ((2 + 3)/2, (6 + (-3))/2) = (5/2, 3/2).
  4. Let B have coordinates (x, y) and find the midpoint of OB: Midpoint of OB = ((0 + x)/2, (0 + y)/2) = (x/2, y/2).
  5. Equate the midpoints of AC and OB: Since the diagonals bisect each other, (x/2, y/2) = (5/2, 3/2).
  6. Solve for x: x/2 = 5/2 => x = 5.
  7. Solve for y: y/2 = 3/2 => y = 3.
  8. State the coordinates of B: The coordinates of B are (5, 3).

This step-by-step breakdown highlights the logical progression of the solution, emphasizing the importance of each step in arriving at the correct answer. By carefully applying the midpoint formula and leveraging the properties of parallelograms, we have successfully determined the coordinates of point B. This methodical approach can be applied to other geometric problems, reinforcing the value of structured problem-solving.

Verification of the Solution

To ensure the accuracy of our solution, it's crucial to verify that the coordinates of point B, which we found to be (5, 3), indeed form a parallelogram OABC with the given points O(0,0), A(2,6), and C(3,-3). We can verify this by checking if the opposite sides are parallel and equal in length. First, let's consider the sides OA and BC. The slope of OA is (6-0)/(2-0) = 3. To find the slope of BC, we use the coordinates B(5,3) and C(3,-3): Slope of BC = (-3-3)/(3-5) = -6/-2 = 3. Since the slopes of OA and BC are equal, they are parallel. Next, let's calculate the lengths of OA and BC using the distance formula: Distance = √((x₂ - x₁)² + (y₂ - y₁)²). Length of OA = √((2-0)² + (6-0)²) = √(4 + 36) = √40. Length of BC = √((3-5)² + (-3-3)²) = √(4 + 36) = √40. Since the lengths of OA and BC are equal, we have one pair of opposite sides that are parallel and equal. Now, let's consider the sides AB and OC. The slope of AB = (3-6)/(5-2) = -3/3 = -1. The slope of OC = (-3-0)/(3-0) = -3/3 = -1. Since the slopes of AB and OC are equal, they are parallel. Let's calculate the lengths of AB and OC: Length of AB = √((5-2)² + (3-6)²) = √(9 + 9) = √18. Length of OC = √((3-0)² + (-3-0)²) = √(9 + 9) = √18. Since the lengths of AB and OC are equal, we have another pair of opposite sides that are parallel and equal. Since both pairs of opposite sides are parallel and equal in length, we can confidently conclude that OABC is indeed a parallelogram, and our solution B(5, 3) is correct. This verification process underscores the importance of checking solutions in mathematical problem-solving, ensuring accuracy and reinforcing understanding of the concepts involved.

Conclusion

In conclusion, we have successfully determined the coordinates of point B in the parallelogram OABC, given the coordinates of points A and C. By leveraging the key property that the diagonals of a parallelogram bisect each other, and by applying the midpoint formula from coordinate geometry, we found the coordinates of B to be (5, 3). This problem serves as a valuable illustration of how geometric properties and algebraic techniques can be combined to solve mathematical problems. The step-by-step approach, from understanding the problem statement to verifying the solution, highlights the importance of structured problem-solving. The verification step further reinforces the accuracy of our result and deepens our understanding of parallelograms and coordinate geometry. This exercise not only enhances our problem-solving skills but also underscores the interconnectedness of different mathematical concepts. The ability to apply these concepts in a cohesive manner is crucial for success in mathematics and related fields. The strategic use of geometric properties and algebraic tools, as demonstrated in this problem, forms a powerful approach to tackling a wide range of mathematical challenges. This exploration reinforces the beauty and utility of mathematics in solving real-world problems and advancing our understanding of the world around us.