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Chapter 3: Problem 20

State the quadrant in which the given point lies. (7,-8)

Short Answer

Expert verified
The point (7, -8) lies in the fourth quadrant.

Step by step solution

01

Review the Coordinate System

The coordinate system is divided into four quadrants. The first quadrant is where both x and y are positive. The second quadrant is where x is negative and y is positive. The third quadrant is where both x and y are negative. The fourth quadrant is where x is positive and y is negative.
02

Analyze the Point (7, -8)

For the point (7, -8), the x-coordinate is 7 and the y-coordinate is -8. Evaluate the signs: x = 7 is positive, y = -8 is negative.
03

Determine the Quadrant

Based on the previous analysis, where the x-coordinate is positive and the y-coordinate is negative, conclude that the point (7, -8) is located in the fourth quadrant of the coordinate system.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Quadrants
In geometry, the coordinate system is a framework used to identify the position of points in a plane. A crucial part of this system is the division into quadrants. These quadrants help us understand where a point is located relative to the origin.

The Cartesian coordinate system divides the plane into four distinct quadrants. These quadrants are labeled counter-clockwise starting from the upper right section:
  • First Quadrant: Both the x and y coordinates are positive.
  • Second Quadrant: The x coordinate is negative, and the y coordinate is positive.
  • Third Quadrant: Both the x and y coordinates are negative.
  • Fourth Quadrant: The x coordinate is positive, and the y coordinate is negative.
Understanding which quadrant a point lies in helps to quickly determine the signs of its x and y coordinates.
X-Coordinate
The x-coordinate of a point in the coordinate system is its horizontal distance from the origin. The x-axis extends horizontally on the plane, and the x-coordinate signifies how far a point is along this axis.

A few key points to remember about the x-coordinate:
  • If the x-coordinate is positive, the point lies to the right of the origin.
  • If the x-coordinate is negative, the point lies to the left of the origin.
  • The x-coordinate is always the first number in a set of coordinates, usually expressed as \(x, y\).
In the example point (7, -8), the x-coordinate is 7, indicating it is located 7 units to the right of the origin.
Y-Coordinate
Just like the x-coordinate, the y-coordinate determines a point's position in the coordinate system, but in the vertical direction. It measures how far up or down a point is from the horizontal x-axis.

Important things about the y-coordinate:
  • If the y-coordinate is positive, the point is above the x-axis.
  • If the y-coordinate is negative, the point is below the x-axis.
  • The y-coordinate is the second number in the coordinate pair, appearing as \(x, y\).
For the specific point (7, -8), the y-coordinate is -8, meaning it is positioned 8 units below the x-axis.
Positive and Negative Signs
Understanding positive and negative signs is essential for pinpointing exact locations within the coordinate grid. These signs indicate the direction and quadrant in which a point resides.

Here's how signs work in the coordinate system:
  • A positive x-coordinate implies the point is right of the y-axis, while a negative x-coordinate means it is left of the y-axis.
  • A positive y-coordinate indicates the point is above the x-axis, and a negative y-coordinate indicates it is below the x-axis.

For instance, the point (7, -8) has a positive x-coordinate and a negative y-coordinate. This combination places it in the fourth quadrant, where x is positive, and y is negative.

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