Quadrants in Geometry

Quadrant is defined as a region in space that is divided into four equal parts by two axes namely the X-axis and the Y-axis in the Cartesian Plane. These two axes intersect each other at 90 degrees and the four regions so formed are called four quadrants, namely I quadrant, II quadrant, III quadrant, and IV quadrant.

In this article, we will explore essential quadrant concepts, including what a quadrant is, its area, quadrant graph, Cartesian Plane, sign conventions within the quadrant, abscissa, and coordinate, as well as the plotting of points on a quadrant.

What are Quadrants of Graph?

A quadrant is a designated area on a Cartesian plane, created by the intersection of the X and Y axes. In this plane, four quadrants are formed, each with its unique traits. The first quadrant, in the upper right, has positive x and y coordinates. The second quadrant, in the upper left, has negative x and positive y coordinates, and so on. Understanding these quadrants is essential for locating and interpreting points on the graph, providing a systematic way to navigate and analyze Cartesian coordinates.

4 Quadrants on Coordinate Plane

The Cartesian plane, formed by the X and Y axes, is split into four quadrants, each with distinct characteristics:

• First Quadrant: Located in the upper right, both x and y-coordinates are positive. This quadrant represents points in the top-right portion of the plane.
• Second Quadrant: Situated in the upper left, the x-coordinate is negative, and the y-coordinate is positive. This quadrant covers points in the top-left part of the plane.
• Third Quadrant: Positioned in the lower left, both x and y-coordinates are negative. Points in the bottom-left area of the plane fall into this quadrant.
• Fourth Quadrant: Found in the lower right, the x-coordinate is positive, and the y-coordinate is negative. This quadrant includes points in the bottom-right portion of the plane.

The quadrants are numbered in an anti-clockwise direction, starting from the upper right. The point where the X and Y axes intersect is called the origin, with coordinates (0,0), indicating zero values for both x and y. Understanding these quadrants helps locate points within the Cartesian plane.

What is Origin?

Starting point on a graph, known as the origin and shown as (0, 0), is where the horizontal x-axis and the vertical y-axis intersect. This means that at the origin, the values for both x and y are zero. It serves as a reference point for locating other points on the graph. In the image added above point O shows the origin.

Abscissa and Ordinate in Quadrants

In the four quadrants, numbers are represented as pairs (a, b), where ‘a’ stands for the x-coordinate, and ‘b’ for the y-coordinate. To figure out where a point is without plotting, pay attention to the signs of the x-coordinate (abscissa) and y-coordinate (ordinate). For example, if you have a point like Q (3, -5), the signs (+ve, -ve) indicate it’s in quadrant IV.

The abscissa shows the horizontal distance from the Y-axis. A positive abscissa means to the right, and in our example, abscissa = 3 means go right from the origin along the x-axis by 3 units.

The ordinate indicates the vertical distance from the origin. A negative ordinate means to go down from the origin along the y-axis. In the example, ordinate = -5 means go down by 5 units.

Sign Convention in Quadrants

Sign conventions in the quadrants can be easily understood using the image added below,

In the XY plane, as we move from left to right along the x-axis, the x-coordinate increases. Similarly, along the y-axis, moving from bottom to top results in an increase in the y-coordinate. The XY plane is divided into four quadrants, each with specific sign conventions for x and y coordinates:

Quadrant	x-coordinate	y-coordinate
1st Quadrant	Positive (+)	Positive (+)
2nd Quadrant	Negative (-)	Positive (+)
3rd Quadrant	Negative (-)	Negative (-)
4th Quadrant	Positive (+)	Negative (-)

Therefore, points in the 1st quadrant have positive values for both x and y, those in the 2nd quadrant have a negative x and a positive y, the 3rd quadrant has both negative x and y values, and the 4th quadrant has a positive x and a negative y.

Plotting Points on Quadrants

In a Cartesian plane, points are identified by the x-axis and y-axis. These points are denoted as (a, b), where ‘a’ is the x-coordinate (abscissa), and ‘b’ is the y-coordinate (ordinate). To position a point in a quadrant, we consider the signs of these coordinates. The values of x and y represent how far the point is from the x-axis and y-axis, respectively.

For example, plot the point (2, -5) on the Cartesian plane. Analyzing the sign of the coordinates reveals that the point is in the 4th quadrant. It will be 2 units away from the x-axis (to the right) and 5 units away from the y-axis (down), using the origin as a reference point.

Trigonometric Values in Different Quadrants

The values of various trigonometric functions in different quadrants can be learn by studying the table added below as,

Quadrant	Sin	Cos	Tan	Cosecant	Secant	Cotangent
1st Quadrant	+	+	+	+	+	+
2nd Quadrant	+	–	+	+	–	–
3rd Quadrant	–	–	–	–	–	+
4th Quadrant	–	–	–	–	+	–

In the 1st quadrant, all trigonometric ratios are positive. In the 2nd quadrant, Sine and Cosecant are positive (+), while Cosine and Secant are negative (-). In the 3rd quadrant, Tangent and Cotangent are positive (+), while Cosine and Secant are negative (-). In the 4th quadrant, Sine and Cosecant are negative (-), while Cosine and Secant are positive (+).

Read More,

• Coordinate Geometry
• Parallel Lines
• Distance Formula

Solved Examples on Quadrant

Example 1: Plot the point A (3, -4) and identify its Quadrant.

Solution:

Point A is located at coordinates (3, -4). Since the x-coordinate is positive (3) and the y-coordinate is negative (-4), Point A lies in Quadrant IV.

Example 2: Plot the point P (-5, 2) and determine its quadrant

Solution:

Coordinates of point P are (-5, 2). To determine the quadrant, we examine the signs of the x and y coordinates.

X-coordinate is -5, indicating a position to the left of the origin.

Y-coordinate is 2, indicating a position above the origin.

Therefore, since the x-coordinate is negative and the y-coordinate is positive, point P is located in Quadrant II.

Point P (-5, 2) is situated in Quadrant II of the Cartesian plane.

Practice Problems on Quadrants

Problem 1: Plot the point (1, -1) and identify its quadrant.

Problem 2: Find three points on the x-axis and determine their quadrants.

Problem 3: If a point lies on the y-axis with coordinates (0, -3), which quadrant is it in?

Problem 4: Locate the points Q (2, 2), R (-2, -2), and S (0, 0) and check for collinearity.

Problem 5: Plot the point (-4, -3) and explain in which quadrant it is situated.

FAQs on Quadrants

1. What is a Quadrant in Maths?

In mathematics, a quadrant is one of the four sections created by the intersection of two perpendicular lines or axes. These axes are typically labeled as the x-axis and y-axis in a Cartesian coordinate system.

2. What is the Intersection of Two Axes Called?

The intersection of two axes in a Cartesian coordinate system is called the origin. It is represented by the point where the x-axis and y-axis meet, usually denoted as (0,0).

3. What are 4 Quadrants?

The four quadrants are the sections formed by dividing a Cartesian coordinate plane into four equal parts. They are labeled as the first quadrant (Q1), second quadrant (Q2), third quadrant (Q3), and fourth quadrant (Q4).

4. Which Quadrant is Positive?

The positive quadrant in a Cartesian coordinate system is the first quadrant (Q1). In this quadrant, both the x and y coordinates are positive.

5. What is the Use of Quadrants in Graphs?

Quadrants in graphs provide a systematic way to organize and locate points based on their coordinates. They help in visualizing relationships between variables and analyzing patterns in data sets, making it easier to interpret graphical representations.

6. Which Quadrant has both the values of Coordinates Positive?

The first quadrant (Q1) is the quadrant where both the x and y coordinates of points are positive. It is the only quadrant with both positive values.

7. What are the 4 Quadrants of a Circle?

The concept of quadrants is not directly applicable to circles. Instead, circles are divided into angles measured in degrees. However, if referring to circular sectors, one might use terms like first sector, second sector, third sector, and fourth sector, corresponding to different angular regions.