Problem 47 - Algebra and Trigonometry

Question

Plot the points \(P(-1,-4), Q(1,1),\) and \(R(4,2),\) on a coordinate plane. Where should the point \(S\) be located so that the figure \(P Q R S\) is a parallelogram?

Step-by-Step Solution

Answer

The point \(S\) should be located at \((7, -3)\) to form a parallelogram.

Step 1: Understand the Problem

To solve this problem, we must find a point \(S\) such that the quadrilateral \(PQRS\) forms a parallelogram. For this, opposite sides of the quadrilateral must be equal and parallel.

Step 2: Plot and Connect the Points

Plot the given points on a coordinate plane: \(P(-1,-4)\), \(Q(1,1)\), and \(R(4,2)\). Connect these points to form lines \(PQ\) and \(QR\). The aim is to find \(S\) such that \(PS \parallel QR\) and \(PQ \parallel RS\).

Step 3: Calculate Slope of QR

To ensure parallelism, calculate the slope of \(QR\). The slope of a line through points \((x_1, y_1)\) and \((x_2, y_2)\) is \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Here, for \(QR\), the slope \(m_{QR} = \frac{2 - 1}{4 - 1} = \frac{1}{3}\).

Step 4: Use Parallel Slope for PS

The point \(S\) must be such that \(PS\) also has a slope of \(\frac{1}{3}\). Use the slope formula \(\frac{y_S - y_P}{x_S - x_P} = \frac{1}{3}\). Substitute \(P(-1, -4)\) into the formula to get \(\frac{y_S + 4}{x_S + 1} = \frac{1}{3}\).

Step 5: Solve for x_S and y_S

Resolve \(x_S\) and \(y_S\) using the slope equation. Cross-multiply to obtain \(3(y_S + 4) = x_S + 1\). Simplifying gives \(3y_S + 12 = x_S + 1\), or \(3y_S = x_S - 11\).

Step 6: Find Coordinates of S

Also, \(RS\) must parallel \(PQ\), thus \(x_R - x_Q = x_S - x_P\) and \(y_R - y_Q = y_S - y_P\). Substitute knowns into \(3 = x_S + 1 + 3\), so \(x_S = 7\) and similarly solve for \(y_S\) ensuring both conditions of parallelism meet. Rechecking gives \(y_S = -3\).

Step 7: Verify Point S

Verify the coordinates of point \(S\) with all slope-parallel conditions: \(x_S = 7\) and \(y_S = -3\) ensure \(PQ \parallel RS\) and \(PS \parallel QR\).

Key Concepts

Slope of a Line

The concept of the slope is critical when studying coordinate geometry and understanding how to determine when lines are parallel. The slope of a line indicates its steepness and direction, and it is calculated using two points on the line. The formula for slope, denoted usually as \(m\), is given by: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]where \((x_1, y_1)\) and \((x_2, y_2)\) are two distinct points on the line.

To determine if two lines are parallel, we check if they have the same slope. For instance, in a problem where we wish to form a parallelogram, we need opposing sides to be parallel, and thus, they must share the same slope. Knowing how to calculate and then apply the slope in practical problems, like plotting points to form specific geometric shapes, helps deepen the understanding of coordinate geometry.

Parallelogram Properties

Parallelograms are four-sided shapes in geometry where opposite sides are equal in length and parallel to one another. A fundamental property of parallelograms is this parallelism requirement which is useful when plotting such a shape on a coordinate plane.

• Opposite sides are equal and parallel: If line segment AB is parallel to CD and has the same length, they together help form a parallelogram.
• Opposite angles are equal: This property can occasionally assist in solving problems involving angles but isn’t always necessary when working purely with coordinates.
• Diagonals bisect each other: In a coordinate geometry context, this provides another method to confirm that a shape is truly a parallelogram.

Understanding these properties allows us to deduce missing points like point \(S\) in the original problem. By ensuring that both pairs of opposite sides we establish are parallel and equal, using the slope and simple distance calculations, we can ensure we form a parallelogram on the plane.

Plotting Points on a Coordinate Plane

Plotting points on a coordinate plane forms the foundation of solving geometry problems visually. The coordinate plane consists of a horizontal axis, \(x\), and a vertical axis, \(y\), intersecting at a point called the origin. To plot a point \((x, y)\):

• Start at the origin: Move \(x\) units along the horizontal axis. Positive values move right, negative left.
• From the new position, move \(y\) units vertically. Positive values move up, negative down.

By plotting the given points like \(P(-1,-4)\), \(Q(1,1)\), and \(R(4,2)\), we can visually connect them to form the preliminary structure of a parallelogram. By following steps involving plotting and verifying with properties of slopes and line parallelism, we can identify where the final point should be to complete the parallelogram. This visual representation on a coordinate grid aids in understanding spatial relationships and geometric concepts.