Question

Graph each set of points, connect them, and identify the geometric figure formed. \(\left(2,-\frac{1}{2}\right),\left(3,-1 \frac{1}{2}\right),\left(1 \frac{1}{2},-3\right),\) and \(\left(\frac{1}{2},-2\right)\)

Short answer

After plotting the points and connecting them in order, the geometric figure formed appears to be an irregular quadrilateral.

Step-by-step solution

Plotting the Points

Begin by plotting each of the given points on a coordinate plane. Plot the point (2, -1/2) on the graph; then plot the point (3, -1.5) on the graph; plot the point (1.5, -3) on the graph, and finally plot the point (0.5, -2) on the graph.

Connecting the Points

After all points are plotted on the coordinate plane, draw straight lines to connect them in the order they were given. Connect the point (2, -1/2) to (3, -1.5), then (3, -1.5) to (1.5, -3), then (1.5, -3) to (0.5, -2), and finally, connect (0.5, -2) back to (2, -1/2) to close the shape.

Identifying the Geometric Figure

Observe the shape formed by connecting the points. The vertices should create a closed figure. Determine the type of geometric figure by the number of sides (four sides would indicate a quadrilateral), the lengths of the sides, and the measures of the angles (if possible).

Key concepts

Graphing Geometric Figures

Graphing geometric figures on a coordinate plane is a fundamental skill in understanding the relationship between algebra and geometry. When you plot points given as pairs of numbers known as coordinates, you are establishing a visual representation of mathematical concepts. The first number, or the x-coordinate, indicates the horizontal position, while the second number, the y-coordinate, indicates the vertical position.

To graph a geometric figure, start with plotting the individual points. Each point corresponds to a vertex of your figure. In our exercise, you plot the points \( (2,-\frac{1}{2}), (3,-1\frac{1}{2}), (1\frac{1}{2},-3), \) and \( (\frac{1}{2},-2) \) on a coordinate grid. After plotting, you then connect these points in the given order with straight lines to form the figure.

Remember, the accuracy in plotting ensures that the figure you form is a true representation of the intended shape. When all points are connected, be sure to close the shape by returning to the start point, forming a loop. This step makes your geometric figure recognizable and complete.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a bridge between algebra and geometry where geometric figures are analyzed and represented using coordinates and equations. The coordinate plane itself consists of two perpendicular lines: one horizontal (the x-axis) and one vertical (the y-axis). These axes intersect at a point called the origin, which has the coordinates \( (0,0) \).

Solving geometric problems in coordinate geometry often involves plotting points, calculating distances between points, and determining the slopes of lines. When you're graphing, every point on the plane is identified by an \( (x, y) \) coordinate pair, which denotes its position relative to the origin.

In our example, you use the given coordinates to determine the position of each point. It's key to ensure that you read the coordinates correctly and place each point in the exact location appropriate to its values along the axes. This precision is what allows for accurate graphing of geometric figures and is essential in solving more complex problems in coordinate geometry.

Identifying Polygons

Identifying polygons on the coordinate plane involves recognizing the characteristics that define various polygon types, such as triangles, quadrilaterals, pentagons, and so forth. A polygon is a closed geometric figure with straight sides that are fully connected. To identify the type of polygon, count the number of vertices (corners) or sides the shape has.

In the context of our exercise, after connecting the points in the given order, you should examine the resulting figure to determine what polygon it represents. Consider factors like the number of sides, which should match the number of vertices, and whether the sides are of equal length, which can indicate regular polygons. In some cases, you may also be able to deduce the measure of angles to help differentiate between shapes like rectangles and parallelograms.

Moreover, understanding how to classify polygons can be helpful for future endeavors in geometry, as certain properties and formulas apply only to specific types of polygons. When practicing these skills, you enhance your ability to visualize and solve more intricate problems in geometry.