Problem 13 - Technical Mathematics with Calculus

Question

Graph each set of points, connect them, and identify the geometric figure formed. \((0.7,2.1),(2.3,2.1),(2.3,0.5),\) and (0.7,0.5)

Step-by-Step Solution

Answer

After plotting the points and connecting them in order, a rectangle is formed.

Step 1: Plot the Points

Begin by plotting each of the given points on a coordinate grid. Plot point A at (0.7, 2.1), plot point B at (2.3, 2.1), plot point C at (2.3, 0.5), and plot point D at (0.7, 0.5). Make sure that each point is accurately placed according to its x (horizontal) and y (vertical) values.

Step 2: Connect the Points

Draw straight lines to connect the points in the order they were given. Connect point A to point B, point B to point C, point C to point D, and finally point D back to point A to close the shape.

Step 3: Identify the Geometric Figure

Examine the shape formed by connecting the points. The opposite sides are equal in length (A to B is the same length as C to D, and B to C is the same length as D to A) and parallel (AB is parallel to CD, BC is parallel to AD), indicating that the shape is a rectangle.

Key Concepts

Coordinate Grid

In mathematics, the coordinate grid is an essential tool for graphing and visualizing the relationship between sets of numbers. It is a two-dimensional plane formed by the intersection of two perpendicular number lines: the horizontal line known as the x-axis and the vertical line known as the y-axis. Each axis is scaled with evenly spaced marks, and every point on the plane can be identified by an ordered pair of numbers, or coordinates, which indicates its precise location.To plot a point, like (0.7, 2.1), you start at the origin—the point where the x-axis and y-axis intersect—and move 0.7 units right along the x-axis and then 2.1 units up along the y-axis. This process is repeated for every point you wish to plot. A clear understanding of how to navigate this system is crucial when plotting points or drawing geometric figures.

Geometric Figures

Geometric figures are shapes or forms that consist of points and lines, which can exist in two or three dimensions. In the context of a coordinate grid, we are usually dealing with two-dimensional shapes like triangles, squares, circles, and more complex polygons. The position and relation of the points on the grid determine the type of geometric figure formed. For instance, when you connect plotted points in a certain order and find that the sides and angles meet specific criteria, you can identify the figure as a certain shape, like the rectangle in the exercise. Careful analysis of properties such as side length and angle measurement are key in determining the exact nature of the geometric figure.

Rectangles

Rectangles are four-sided geometric figures with some very specific properties. First and foremost, they have four right angles, meaning the angles between adjacent sides are all 90 degrees. Additionally, opposite sides of a rectangle are not only equal in length but also parallel, which is a function of those right angles maintaining consistency across the shape.When identifying rectangles on a coordinate grid, like the one in the provided exercise, we look for these properties. Once you've plotted the points and connected them to form a shape, ensure that the opposing sides are of equal length and parallel. If both these conditions are satisfied, the geometric figure you're examining can be confidently identified as a rectangle.