Problem 18 - Algebra and Trigonometry

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 either (10,6) or (-5,6). The area of the square is 100 square units.

Step 1: Plot Points P, Q, and R

Determine the location of each point on the coordinate plane. Point \( P(5,1) \) is at 5 on the x-axis and 1 on the y-axis. Point \( Q(0,6) \) is at 0 on the x-axis and 6 on the y-axis. Point \( R(-5,1) \) is at -5 on the x-axis and 1 on the y-axis. Plot these points on the coordinate grid.

Step 2: Determine Potential Position of Point S

Since the quadrilateral must be a square, each side must be equal in length. Observing that both points \( P \) and \( R \) have a y-coordinate of 1, S must also lie on this horizontal line to satisfy symmetry, making \( PS \) and \( RS \) equal.

Step 3: Calculate Side Length

Since the square must have all sides of equal length, examine the distance between \( P \) and \( R \). We can calculate this using the distance formula: \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \). Plugging in the values, between \( P \) and \( R \): \( \sqrt{(-5 - 5)^2 + (1 - 1)^2} = \sqrt{100} = 10 \). Thus, each side of the square is 10 units in length.

Step 4: Determine Coordinates of Point S

To form a square, \( S \) can be located on two possible positions based on side lengths: it either extends horizontally from \( Q \) or vertically. If we choose the horizontal extension, we calculate assuming a horizontal distance of 10 units (the same as \( PQ \)). So \( S(0 + 10, 6) = (10, 6) \). Alternatively, if we choose vertical extension keeping same side length from Q, \( S \) coordinates will be \( (-5,6) \).

Step 5: Calculate Area of the Square

Using the side length calculated, the area \( A \) of the square is \( s^2 \) where \( s \) is the side length. Since we previously found the side length to be 10, the area is \( 10^2 = 100 \) square units.

Key Concepts

Coordinate plane

The coordinate plane is a two-dimensional space defined by a horizontal line called the x-axis and a vertical line called the y-axis. These axes intersect at a point called the origin, which has coordinates (0, 0). Every point on the plane can be described by a pair of numerical coordinates

• The first number represents its position along the x-axis.
• The second number represents its position along the y-axis.

To visualize this, imagine a grid where points like P(5,1), Q(0,6), and R(-5,1) are located by finding 5 units right and 1 unit up for P, 0 units horizontally and 6 units up for Q, and 5 units left and 1 unit up for R. This grid is essential for solving problems related to positioning, distance, and geometry.

Distance formula

The distance formula is a tool used to calculate the distance between two points on the coordinate plane. It's derived from the Pythagorean theorem and is given by:\[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]Here,

• \( (x_1, y_1) \) and \( (x_2, y_2) \) are the coordinates of the two points.
• \( d \) is the distance between them.

For example, to find the distance between P(5,1) and R(-5,1), you use their x-coordinates and y-coordinates in the formula: \[d = \sqrt{(-5 - 5)^2 + (1 - 1)^2} = \sqrt{100} = 10\]This tells us that points P and R are 10 units apart, which is crucial when determining side lengths in geometry problems.

Quadrilateral

A quadrilateral is a polygon with four sides and four vertices. In coordinate geometry, quadrilaterals can be categorized by calculating side lengths and comparing angles. For example, if all sides are equal and angles are right angles, the quadrilateral is a square. Other types include rectangles, rhombuses, and kites. To investigate whether a set of points creates a quadrilateral like a square, consider side lengths and how they intersect. For quadrilateral PQRS to be a square, its sides must equal in length and form right angles. By calculating distances like those between P and R or P and S, you ensure symmetry and correctness in creating a square.

Area of square

The area of a square is defined as the space enclosed within its four sides. This is calculated by squaring the length of one of its sides. The formula for the area is \[A = s^2\]where \( s \) is the length of one side. Understanding this formula helps in determining space and coverage of the structure formed by the coordinates on the plane.In the provided problem, once the side length of the square PQRS was calculated as 10 units, the area could be determined with the formula:\[A = 10^2 = 100\]This means the square covers an area of 100 square units, ensuring all calculations match the geometric requirements of a square.