Problem 22 - Precalculus Mathematics for Calculus

Question

Plot the points \(P(5,1), Q(0,6),\) and \(R(-5,1)\) on a coordinate plane. Where must the point \(S\) be located so that the quadrilateral \(P Q R S\) is a square? Find the area of this square.

Step-by-Step Solution

Answer

Point \( S \) must be at (0,-4) and the area of the square is 100 square units.

Step 1: Plot Initial Points

First, identify and plot the points \( P(5,1) \), \( Q(0,6) \), and \( R(-5,1) \) on the Cartesian coordinate plane. These points will form three vertexes of the quadrilateral.

Step 2: Determine Location of S for a Square

Observe that for quadrilateral \( P Q R S \) to be a square, the sides must be equal, and the angles must be 90 degrees. Plotting \( P, Q, \) and \( R \) reveals they form an "L" shape. The side \( PR \) must be parallel and equal to side \( QS \), and \( QR \) to \( PS \). By calculating, take \( S = (0,-4) \) to complete the square.

Step 3: Calculate Distances to Confirm a Square

The distance \(PQ\) is \( \sqrt{(5-0)^2 + (1-6)^2} = \sqrt{50} \). Similarly, \(QR = \sqrt{50} \) and \(PR = 10\). Calculate \( PS \) and \(QS \) to ensure they are equal to \( QR \) and \( PQ \). \( PS = \sqrt{(5-0)^2 + (1+4)^2} = \sqrt{50} \), and \( QS = \sqrt{(0-0)^2 + (6+4)^2} = 10 \). All sides are either \( \sqrt{50} \) or 10.

Step 4: Calculate the Area of the Square

The square's side length is \(10\). Therefore, use the formula for the area of a square, \( \, \text{Area} = \text{side}^2 \). \( \, \text{Area} = 10^2 = 100 \).

Key Concepts

Coordinate Plane

The coordinate plane is a two-dimensional surface where we can plot points, lines, and shapes using a pair of numbers. Each point is marked by an

• x-coordinate, which shows the horizontal position
• y-coordinate, which shows the vertical position

This system is also known as the Cartesian plane, named after French mathematician René Descartes. The plane is divided into four quadrants by the horizontal x-axis and the vertical y-axis, intersecting at the origin \(0, 0\). Each quadrant tells you the sign of the x and y values. When plotting points in a coordinate plane, you'll always start at the origin and move:

• horizontally first according to the x-value
• then vertically according to the y-value

Distance Formula

The distance formula is a key tool used to determine the distance between any two points on the coordinate plane. It is derived from the Pythagorean theorem. For two points \(A(x_1, y_1)\) and \(B(x_2, y_2)\), the distance \(d\) between them is:\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]This formula provides a way to calculate how far apart the points are, effectively giving us the length of the line segment connecting them. It's especially useful in verifying figures like squares, rectangles, and triangles by comparing their side lengths.

Area Calculation

When calculating the area of shapes like squares, knowing the length of one side is enough. This is because in a square, all sides are equal. The formula for the area \(A\) is:\[ A = \text{side}^2 \]In the exercise, after ensuring that the quadrilateral is a square, its side was calculated to be 10 units. Therefore, the area of the square \(A\) was:\[ A = 10^2 = 100 \]This straightforward calculation confirms the space inside the square in squared units, which is an essential step in many geometry problems.

Points Plotting

Points plotting is the fundamental starting step in working with many geometric problems. It involves placing dots on a coordinate plane at positions given by their coordinates (x, y). For example, in the exercise, you had to plot points \(P(5,1)\), \(Q(0,6)\), and \(R(-5,1)\). To do this manually, consider the following steps:

• First, locate the x-coordinate. Move right for positive values or left for negative ones.
• Second, find the y-coordinate. Move up for positive values or down for negative ones.
• Finally, mark the point where the coordinates meet.

Plotting accurately is crucial for visualizing the shape being analyzed, such as understanding the square configuration in this problem.