What is a Point?
A point is a dot and it is a primitive concept in classical Euclidean geometry that represents a precise location in space and has no length, width, or thickness. A point is a component of a set called space in modern mathematics.
- Two perpendicular lines, or axes, labeled like number lines, compensate for a coordinate grid. The x-axis is the term for the horizontal axis. The y-axis is the term for the vertical axis. The origin is the point where the x and y-axes intersect.
- A point in two-dimensional Euclidean space is represented by an ordered pair of numbers (x, y), with the first number representing the horizontal line x-axis and often denoted by x, and the second number representing the vertical line y-axis and often denoted by y.
What is a Coordinate Plane?
The intersection of a horizontal number line and a vertical number line forms a coordinate plane. An ordered pair is a set of numbers used to find a point in a coordinate plane. The x-coordinate represents the horizontal distance from the origin along the x-axis. The y-coordinate represents the vertical distance from the origin along the y-axis.
For example, (5 , 6)
In the above example, The first number 5 is the x-coordinate and it represents the horizontal distance from the origin along the x-axis. The second number 6 is the y-coordinate and it represents the vertical distance from the origin along the y-axis.
What is a Line Segment?
A line segment is a section of a line that is defined by two distinct end points and includes all points on the line between them.
- A closed line segment contains both endpoints, whereas an open line segment does not; a half-open line segment contains only one endpoint.
- A line segment is frequently denoted in geometry by a line above the symbols for the two endpoints (such as \(\overline{AB}\) )
Finding Distances on the Coordinate Plane
The y-coordinates of all points on a horizontal line are the same. The x-coordinates of all points on a vertical line are the same. There are two methods for calculating the distance between two points on a line segment. Those are:
- Counting units
- Subtraction
When two points are on the same horizontal or vertical line, you can find the distance between them by counting units or using subtraction.
For example: Find the distance between points Q and F from the given coordinate plane.
One way of finding: Using count units.
Step 1: Identify the locations of the points: Point Q is located at (3, 5).
Point F is located at (6, 5).
Step 2: Draw a line segment to connect the points.
Step 3: Count horizontal units: There are 3 units between Q and F.
So, the distance between points Q and F is 3.
Another way of finding: Using subtraction.
Points Q and F have the same Y – coordinates. They lie on a horizontal line. Subtract the X – coordinates to find the distance.
6 – 3 = 3
So, the distance between points Q and F is 3.
Drawing a Polygon in a Coordinate Plane
A polygon is a geometric figure with at least three straight edges and at least three vertices.
Vertices of polygons can be represented or drawn using ordered pairs. Plot and connect the vertices of a polygon in a coordinate plane.
Step 1: In the coordinate plane, plot the vertices. Remember that you have an ordered pair (x,y), the first value indicates where the point should be on the x-axis (left-right), and the second value indicates where the point should be on the y-axis (up-down).
Step 2: To create the desired polygon, draw straight lines from one vertex to the next.
Within the shape, no edges of a polygon will intersect with each other.
For example: The vertices of a polygon are A (3, 3), B (5, 4), C (7, 4), D (7, 3). In a coordinate plane, draw the polygon. Then figure out the type of polygon formed.
Step 1: Plot and label the vertices.
Step 2: Draw the line segments \(\overline{AB},\overline{BC},\overline{CD},\overline{AD}\) and connect the points. We have to be sure to connect the points in order to draw polygons.
Polygon ABCD is an Irregular polygon.
Solved Examples
Example 1: Write the ordered pair that corresponds to point C.
Solution:
Let’s look at the plane. The horizontal distance from the origin to point C is 3 units. So that the x – coordinate is 3.
Now, Again looking at the plane. The vertical distance from the origin to point C is 5 units. So that means the y – coordinate is 5. The ordered pair is (3, 5).
Example 2: Write the ordered pair that corresponds to the point F.
Solution:
Let’s look at the plane. The horizontal distance from the origin to point F is 0 units. So that means the x – coordinate is 0.
Now, again looking at the plane., The vertical distance from the origin to point C is 3 units. So that means the y – coordinate is 3. The ordered pair is (0, 3)
Example 3: Point A is located at ( 1, 2). Plot and label the point.
Solution: Start at the origin and move 1 unit to the right and 2 units to the top. Then plot and label the point on the coordinate plane.
Example 4: Point E is located at ( 3, 5). Plot and label the point.
Solution: Start at the origin and move 3 units to the right and 5 units to the top. Then plot and label the point on the coordinate plane.
Example 5: Find the distance between points M and E from the given coordinate plane.
Solution:
One way of finding: Count units.
Step 1: Identify the locations of the points: Point M is located at (0, 8).
Point E is located at (5, 8).
Step 2: Draw a line segment to connect the points.
Step 3: Count horizontal units: There are 5 units between M and E.
So, the distance between points M and E is 5.
Another way of finding: Using subtraction.
Points M and E have the same Y – coordinates. They lie on a horizontal line. Subtracting the X – coordinates to find the distance.
5 – 0 = 5
So, the distance between points M and E is 5.
Example 6: Newton plotted the points A (3, 7) and B (5, 7) and connected them with a line segment. Descartes says that (9, 7) also lies on the line segment. Is he correct? explain.
Solution: The line segment is in between A (3, 7) and B (5, 7). Now to find whether the C (9, 7) lies within the line segment we have to plot points A (3, 7), B (5, 7), and C (9, 7) and check.
Clearly from the above figure, we can see that point C is not lying on the line segment AB. Therefore point C (9, 7) does not lie on the line segment. So, Descartes is wrong.
Example 7: The vertices of a polygon are A (2, 1), B (2, 4), C (4, 4), and D (4, 1). In a coordinate plane, draw the polygon. Then figure out what it is.
Solution:
Step 1: Plot and label the vertices.
Step 2: Draw the line segments \(\overline{AB},\overline{BC},\overline{CD},\overline{AD}\) and connect the points. We have to be sure to connect the points in the given order to draw a polygon.
Clearly, we can see that polygon ABCD is a rectangle.
Example 8: You must walk 4 blocks east and 3 blocks north to get from the school to the arcade. Your friend walks 2 blocks east and twice as many blocks north to get from school to the skate park. Place the arcade and skate park on a map and label them.
Solution: North means x-axis and east means y-axis.
From the school to the arcade, you walk 4 blocks east and 3 blocks north = (4, 3)
From the school to the skate park, your friend walks 2 blocks east and twice as many blocks north as you = (2, 3\(\times\)2) = (2, 6).
Vertices are individual dots or points. Their x and y – coordinates define them in 2D, and their x, y, and z – coordinates define them in 3D. Connecting these vertices together in order will make the polygon construction easier and you can trace the type of polygon/ shape of polygon easily.
Each point is identified by an ordered pair of numbers: an x – coordinate on the x-axis and a y – coordinate on the y-axis. The ordering is very important and plays a role in computing the normal direction of the coordinates in the plane. If the x – coordinate is interchanged with the y – coordinate then the shapes of the polygon will change and the distances between the points in the coordinate plane will come as negative.