Problem 63 - Algebra and Trigonometry

Question

Plot the points \(P(0,3)\) \(Q(2,2),\) and \(R(5,3)\) on a coordinate plane. Where should the point \(S\) be located so that the figure \(P Q R S\) is a parallelogram? Write a brief description of the steps you took and your reasons for taking them.

Step-by-Step Solution

Answer

Point \(S\) should be located at \((3, 4)\) to form the parallelogram.

Step 1: Understand the Parallelogram Properties

A parallelogram is a quadrilateral with opposite sides that are parallel and equal in length. We need to find a location for the point \(S\) so that \(PS\) is parallel and equal to \(QR\), and \(PQ\) is parallel and equal to \(RS\). This ensures that the figure \(P Q R S\) forms a parallelogram.

Step 2: Plot the Points on the Coordinate Plane

Start by plotting the given points \(P(0,3)\), \(Q(2,2)\), and \(R(5,3)\) on the coordinate plane. These points form a triangle currently.

Step 3: Determine the Vector from P to Q

To calculate the vector \( \overrightarrow{PQ} \), subtract the coordinates of \(P\) from the coordinates of \(Q\): \((2-0, 2-3) = (2, -1)\). This vector represents the direction and length from \(P\) to \(Q\).

Step 4: Use Vector for Parallel Side PS

The vector \( \overrightarrow{QR} \) should equal \( \overrightarrow{PS} \). Calculate \( \overrightarrow{QR} \) by subtracting the coordinates of \(Q\) from \(R\): \((5-2, 3-2) = (3, 1)\).

Step 5: Calculate Point S Using Vectors

If \( \overrightarrow{PS} = \overrightarrow{QR} \), then point \(S\) can be found by starting at \(P\), adding the vector \( \overrightarrow{QR} \): \(S = P + (3, 1) = (0+3, 3+1) = (3, 4)\).

Step 6: Verify Parallelogram with Opposite Side PQ

Now verify using \(PQ\) that \(RS\) will be parallel and equal. Calculate the vector \( \overrightarrow{RS} \) by subtracting \(R\) from \(S\): \((3-5, 4-3) = (-2, 1)\), matching the direction and length of \( \overrightarrow{PQ} = (2, -1)\), reaffirming the opposite and equal sides of a parallelogram.

Key Concepts

Coordinate Plane

A coordinate plane is a two-dimensional surface on which points can be plotted. It consists of two perpendicular lines called axes. The horizontal line is known as the x-axis, and the vertical line is called the y-axis. Where they meet is called the origin, represented as (0,0). The coordinate plane is used to locate points through ordered pairs (x, y), where 'x' represents the horizontal position and 'y' represents the vertical position. Each point on the plane is uniquely identified by these coordinates. When working with figures like parallelograms, it's important to accurately plot these points.

• Helps visualize the geometry of the shape.
• Assists in understanding distances and relationships between points.

In our exercise, we plot points P(0,3), Q(2,2), and R(5,3) to initially form a triangle. Understanding these locations is crucial for finding point S to complete the parallelogram.

Vector Addition

Vector addition is a mathematical technique used to determine the resultant vector from two given vectors. It involves adding the corresponding components of the given vectors to arrive at a new vector.In coordinate geometry, vectors are represented as ordered pairs (x, y), indicating direction and magnitude. When working with shapes like parallelograms, vector addition assists in ensuring sides are parallel and equal.

• Useful for ensuring equal length and parallelism in figures.
• Simplifies computations related to directions and magnitudes.

For this exercise, we apply vector addition to find vector \( \overrightarrow{PS} \), using the condition that \( \overrightarrow{PS} = \overrightarrow{QR} \). By adding the vector \( \overrightarrow{QR} = (3, 1) \) to \( P(0,3) \), we determine S is at (3, 4). This vector addition ensures the side PS is parallel and equal to QR.

Properties of Quadrilaterals

Quadrilaterals are four-sided polygons with a variety of types and properties. A parallelogram is one kind of quadrilateral where opposite sides are both parallel and equal in length. Understanding these key characteristics is essential for identifying and solving problems involving parallelograms.

• Opposite sides are equal in length.
• Opposite angles are equal.
• Diagonals bisect each other.

These properties help confirm a quadrilateral's nature as a parallelogram. In our exercise, properties are used to identify the correct location of point S so that PQRS forms a parallelogram. By ensuring vector \( \overrightarrow{PS} \) matches \( \overrightarrow{QR} \), and vector \( \overrightarrow{RS} \) matches \( \overrightarrow{PQ} \), we confirm that both pairs of opposite sides are equal, meeting parallelogram criteria.

Coordinate Geometry

Coordinate geometry, or analytic geometry, is a mathematics field that uses algebra to explore the properties and relations of geometric shapes on a coordinate plane. It allows the calculation of lengths, angles, areas, and other properties of shapes using their coordinates. In the context of parallelograms, coordinate geometry allows us to employ algebraic techniques to ensure shape properties, like parallel and equal sides, are maintained.

• Combines algebra and geometry to enhance problem-solving.
• Works well with known coordinates to find missing points or verify properties.

Using coordinate geometry in our exercise helps establish conditions for S(3,4) by confirming it through vector calculations and parallelogram properties. This ensures PQRS correctly forms a parallelogram through precise calculations using the coordinate plane.