Identifying The Figure Formed By Points P(0, 0), Q(4, 0), R(4, 5), And S(0, 5)

by ADMIN 79 views

Rushabh plotted the points P(0, 0), Q(4, 0), R(4, 5), and S(0, 5) on a graph paper and joined them to form a closed figure. The question at hand is to identify the type of figure obtained. To solve this, we need to analyze the properties of the figure formed by these points. This involves examining the lengths of the sides and the angles between them. By understanding the coordinates of each point and applying basic geometric principles, we can accurately determine whether the figure is a rhombus, a square, or another quadrilateral. This exploration will not only provide the answer but also enhance our understanding of coordinate geometry and figure classification.

Plotting the Points and Visualizing the Figure

To begin, let's plot the points on a coordinate plane. Point P(0, 0) is the origin. Point Q(4, 0) lies on the x-axis, 4 units to the right of the origin. Point R(4, 5) is 4 units to the right and 5 units up from the origin. Finally, point S(0, 5) lies on the y-axis, 5 units above the origin. Connecting these points in the order P-Q-R-S-P reveals a quadrilateral. This visual representation is crucial because it gives us a preliminary idea of the shape we are dealing with. Without plotting, it's harder to grasp the spatial relationships between the points. The initial visual inspection suggests that the figure is either a rectangle or a square, but we need to verify this with mathematical calculations.

  • Understanding Coordinate Geometry: Coordinate geometry provides a powerful way to analyze geometric shapes using algebraic methods. Each point in the plane is represented by an ordered pair (x, y), which specifies its position relative to the origin (0, 0). The x-coordinate represents the horizontal distance from the origin, and the y-coordinate represents the vertical distance. By using these coordinates, we can calculate distances between points, determine slopes of lines, and identify various geometric properties of shapes.
  • Visualizing the Quadrilateral: Visualizing the quadrilateral formed by the given points is essential for understanding its properties. By plotting the points P(0, 0), Q(4, 0), R(4, 5), and S(0, 5) on a coordinate plane, we can see the shape that is created when these points are connected in sequence. The visualization allows us to make an initial assessment of the figure’s type, such as whether it is a rectangle, square, parallelogram, or some other quadrilateral. This step is crucial for guiding our subsequent calculations and analysis.

Calculating the Lengths of the Sides

Now, let's determine the lengths of the sides of the figure. We can use the distance formula, which is derived from the Pythagorean theorem, to calculate the distance between two points in a coordinate plane. The distance d between two points (x1, y1) and (x2, y2) is given by:

d = √((x2 - x1)² + (y2 - y1)²)

Applying this formula to our points:

  • Length of PQ: d = √((4 - 0)² + (0 - 0)²) = √(4² + 0²) = √16 = 4 units
  • Length of QR: d = √((4 - 4)² + (5 - 0)²) = √(0² + 5²) = √25 = 5 units
  • Length of RS: d = √((0 - 4)² + (5 - 5)²) = √((-4)² + 0²) = √16 = 4 units
  • Length of SP: d = √((0 - 0)² + (0 - 5)²) = √(0² + (-5)²) = √25 = 5 units

From these calculations, we observe that PQ = RS = 4 units and QR = SP = 5 units. This indicates that the figure has two pairs of equal sides, which is a characteristic of rectangles and parallelograms. However, to differentiate between these, we need to examine the angles. Calculating the side lengths is a crucial step in identifying the figure, as it helps narrow down the possibilities based on the properties of different quadrilaterals. This quantitative approach provides a more precise understanding of the figure's dimensions and sets the stage for further analysis.

  • Applying the Distance Formula: The distance formula is a fundamental tool in coordinate geometry for calculating the length of a line segment between two points. Derived from the Pythagorean theorem, this formula allows us to find the distance using the coordinates of the points. By applying the distance formula to each pair of points (PQ, QR, RS, and SP), we obtain the lengths of the sides of the figure. This quantitative approach is crucial for determining the shape of the quadrilateral and distinguishing between different types of quadrilaterals.
  • Analyzing Side Lengths: The side lengths of a quadrilateral provide important clues about its nature. If opposite sides are equal in length, the quadrilateral is likely a parallelogram or a rectangle. If all sides are equal, the quadrilateral could be a rhombus or a square. The lengths PQ = 4 units, QR = 5 units, RS = 4 units, and SP = 5 units reveal that the figure has two pairs of equal sides, suggesting it is a rectangle or parallelogram. This analysis helps us narrow down the possibilities and guides our next steps in identifying the figure.

Checking the Angles

To determine whether the figure is a rectangle or just a parallelogram, we need to check the angles. In a rectangle, all angles are right angles (90 degrees). We can check if the angles are right angles by examining the slopes of the lines. If two lines are perpendicular, the product of their slopes is -1. Let's calculate the slopes of the lines:

  • Slope of PQ: m1 = (0 - 0) / (4 - 0) = 0 / 4 = 0
  • Slope of QR: m2 = (5 - 0) / (4 - 4) = 5 / 0 = undefined (vertical line)
  • Slope of RS: m3 = (5 - 5) / (0 - 4) = 0 / -4 = 0
  • Slope of SP: m4 = (0 - 5) / (0 - 0) = -5 / 0 = undefined (vertical line)

The slope of PQ is 0, indicating a horizontal line, and the slope of QR is undefined, indicating a vertical line. Similarly, the slope of RS is 0, and the slope of SP is undefined. Since horizontal and vertical lines are perpendicular, the angles at P, Q, R, and S are all right angles. This confirms that the figure is a rectangle. The angle check is crucial in differentiating between parallelograms and rectangles. While parallelograms have opposite sides equal, rectangles have the added property of having all angles equal to 90 degrees. This step solidifies our understanding of the figure's properties and confirms its classification.

  • Understanding Slopes of Lines: The slope of a line is a measure of its steepness and direction. It is defined as the change in the y-coordinate divided by the change in the x-coordinate between two points on the line. The slope provides valuable information about the line’s orientation. A slope of 0 indicates a horizontal line, while an undefined slope indicates a vertical line. Perpendicular lines have slopes that are negative reciprocals of each other, meaning their product is -1. Understanding the concept of slopes is crucial for determining angles and identifying geometric properties of figures.
  • Verifying Right Angles: To verify that the angles in the quadrilateral are right angles, we need to check the slopes of the adjacent sides. If the product of the slopes of two adjacent sides is -1, then those sides are perpendicular, forming a right angle. By calculating the slopes of PQ, QR, RS, and SP, we found that PQ and QR are perpendicular, as are QR and RS, RS and SP, and SP and PQ. This confirms that all angles in the quadrilateral are 90 degrees, which is a key property of rectangles and squares. This angle verification is a critical step in accurately identifying the figure.

Conclusion: Identifying the Figure

Based on our calculations, we have determined that the figure formed by the points P(0, 0), Q(4, 0), R(4, 5), and S(0, 5) is a rectangle. We found that the opposite sides are equal in length (PQ = RS and QR = SP), and all angles are right angles. This aligns with the definition of a rectangle, which is a quadrilateral with opposite sides of equal length and all angles equal to 90 degrees. The key steps in our analysis included plotting the points, calculating side lengths using the distance formula, and verifying the angles by examining the slopes of the lines. This systematic approach allows us to confidently identify the figure based on its geometric properties. This exercise not only provides the answer but also reinforces the application of coordinate geometry principles in identifying geometric shapes. Understanding how to analyze figures using coordinates and formulas is a fundamental skill in mathematics and has broad applications in various fields.

  • Summary of Findings: To summarize our findings, we plotted the points P(0, 0), Q(4, 0), R(4, 5), and S(0, 5) and connected them to form a quadrilateral. We then used the distance formula to calculate the lengths of the sides, finding that PQ = RS = 4 units and QR = SP = 5 units. This indicated that the figure had two pairs of equal sides. Next, we calculated the slopes of the lines and determined that adjacent sides were perpendicular, meaning all angles were right angles. Based on these properties, we concluded that the figure is a rectangle.
  • Final Identification: The systematic approach of plotting points, calculating side lengths, and verifying angles has led us to the definitive identification of the figure as a rectangle. This conclusion is supported by the figure's properties: equal opposite sides and all right angles. Our analysis demonstrates the power of coordinate geometry in analyzing and identifying geometric shapes. This exercise reinforces the importance of applying mathematical principles to visual and spatial problems.

Therefore, the figure obtained by joining the points P(0, 0), Q(4, 0), R(4, 5), and S(0, 5) is a rectangle, not a rhombus or a square.