Question

Critical Thinking Graph \(A(-3,-2)\) and \(B(2,-2)\). Draw \(\overline{A B}\). Find th coordinates of two other points that when connected with \(A\) and \(B\) would form a 5-by-3 rectangle.

Short answer

The points are \(C(-3,1)\) and \(D(2,1)\).

Step-by-step solution

Understand the Problem

We are given two points, \(A(-3,-2)\) and \(B(2,-2)\), and we are tasked with finding two additional points to form a 5-by-3 rectangle when connected with points \(A\) and \(B\).

Determine the Length of Existing Side

Calculate the distance between points \(A\) and \(B\) which lie on a horizontal line (same \(y\)-coordinates). \(AB\) measures \(5\) units because the difference between the \(x\)-coordinates is \(2 - (-3) = 5\).

Use Rectangle Properties

A rectangle has opposite sides that are equal. Since \(AB\) is \(5\) units and forms one side of the rectangle, the side parallel to \(AB\) will also be \(5\) units. The other two sides must be \(3\) units vertically (since the question specifies a 5-by-3 rectangle).

Calculate Coordinates of the New Points

By adding 3 units vertically (in the \(y\)-direction) to points \(A\) and \(B\), we can find the new points that will create a 5-by-3 rectangle. This gives us the points \(C(-3,1)\) and \(D(2,1)\).

Verify the Rectangle

To ensure the shape is a rectangle, check if \(AC = BD = 3\) and \(CD = AB = 5\). The calculations show that point \(C(-3,1)\) is \(3\) units above \(A(-3,-2)\), and \(D(2,1)\) is \(3\) units above \(B(2,-2)\) confirming \(AC = BD = 3\). Both \(AB\) and \(CD\) have a length of \(5\), confirming the opposite sides are equal.

Key concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a mathematical discipline that utilizes a coordinate system to describe geometric shapes and their properties.

• It brings algebra and geometry together through graphs and plots of algebraic equations.
• Every point in coordinate geometry is defined as a pair of numbers (x, y) called coordinates.
• These coordinates help in defining and exploring geometric shapes such as lines, circles, and polygons within a plane.

The exercise "finding points to form a rectangle" uses coordinate geometry effectively. Points A(-3,-2) and B(2,-2) are already plotted on the coordinate plane, with the task to identify two additional points that, when joined, will help in forming a rectangle. Understanding how to navigate the coordinate plane is essential for solving such problems.
With a solid grasp of coordinate geometry, you deepen your understanding of how algebraic equations interact with geometric forms on the coordinate plane.

Rectangles

A rectangle is a quadrilateral with four right angles. Key properties of rectangles are:

• Opposite sides are equal and parallel.
• Each angle is 90 degrees.
• The diagonals bisect each other and are equal in length.

In the exercise, we wanted to build a rectangle using specific measurements: 5-by-3.
This dimension indicates that one pair of opposite sides should measure 5 units and the other pair should measure 3 units.
Using properties of rectangles helps in deriving two more points, particularly by ensuring that opposite sides maintain equal lengths. Verifying these measurements confirms that the shape formed is indeed a rectangle and not any other type of quadrilateral.

Distance Formula

The distance formula in coordinate geometry is a fundamental tool to calculate the distance between two points on a coordinate plane:\[D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]Here, \(x_1, y_1\) and \(x_2, y_2\) represent the coordinates of the two points.
This formula is crucial in determining the lengths of sides and diagonals in geometric figures.
In the given problem, the horizontal distance was found using a simplified version of the formula because the y-coordinates for points A and B were the same, making the distance calculation straightforward: \(AB = 2 - (-3) = 5\) units.
By understanding and applying the distance formula effectively, students can verify the lengths of sides and also follow up with checking if the diagonals are equal, ensuring the quadrilateral formed is accurately a rectangle.

Critical Thinking

Critical thinking is an essential skill in geometric problem-solving, especially in scenarios requiring the deduction of unknown elements within given constraints.

• It involves analytical reasoning and employing systematic approaches to dissect a problem.
• Engaging in critical thinking prompts students to hypothesize, experiment, and verify conclusions.

In this problem, critical thinking is utilized to deduce the missing points that complete the rectangle. One must consider the given measurements, understand the properties of a rectangle, and apply these logically to derive solutions.
By going beyond rote calculation, critical thinkers ask themselves questions such as "What other properties can verify this shape is a rectangle?" or "How do the distances reinforce the structural integrity of a rectangle?"
Emphasizing critical thinking, students broaden their approach toward not just finding a solution but understanding the reasoning that guides them there, which in turn reinforces their overall mathematical competency.