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#### Learning Outcomes
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1. Know the basics of the rectangular coordinate system and the quadrants.
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2. Recognize, show, or plot a point on the coordinate plane.
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2. Recognize, show, or plot a point on the coordinate plane.
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## 1️⃣ Basics of the Rectangular Coordinate System and Quadrants
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The **rectangular coordinate system** (also called the Cartesian coordinate system) uses two number lines that intersect at right angles at a point called the **origin (0,0)**.
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- The **horizontal axis** is the \$ x \$-axis.
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- The **vertical axis** is the \$ y \$-axis.
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These axes divide the plane into **four quadrants**, numbered *counterclockwise* as follows:
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-**Quadrant I:**\$ x > 0, y > 0 \$ (top right)
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-**Quadrant II:**\$ x < 0, y > 0 \$ (top left)
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-**Quadrant III:**\$ x < 0, y < 0 \$ (bottom left)
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-**Quadrant IV:**\$ x > 0, y < 0 \$ (bottom right)
Hello! On this Wednesday evening here in India, I would be happy to help you with these questions on the rectangular coordinate system. Let's break them down one by one.
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{{< border >}}
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### **Question 1: Locating Points**
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**The Question:**
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Choose the correct option with respect to the points $P(5, -3)$, $Q(-3, 3)$, $R(0, -100)$, and $S(-2.5, 0)$ on the rectangular coordinate system.
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* Point R does not lie in any quadrant
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* Points P and R lie in Quadrant III
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* Points S and Q lie in Quadrant II
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* Points R and S cannot be represented on the rectangular coordinate system
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**Core Concepts: Quadrants and Axes**
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To solve this, we need to know where points are located based on the signs of their x and y coordinates.
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***Quadrant I:** x is positive (+), y is positive (+)
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***Quadrant II:** x is negative (-), y is positive (+)
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***Quadrant III:** x is negative (-), y is negative (-)
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***Quadrant IV:** x is positive (+), y is negative (-)
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***On an Axis:** If either the x or y coordinate is 0, the point is not in a quadrant but lies on one of the axes.
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* If x = 0, the point is on the y-axis.
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* If y = 0, the point is on the x-axis.
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**Detailed Solution:**
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Let's analyze the location of each point:
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1.**P(5, -3):** The x-coordinate (5) is positive, and the y-coordinate (-3) is negative. A (+, -) point lies in **Quadrant IV**.
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2.**Q(-3, 3):** The x-coordinate (-3) is negative, and the y-coordinate (3) is positive. A (-, +) point lies in **Quadrant II**.
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3.**R(0, -100):** The x-coordinate is 0. This means the point lies directly on the **y-axis**. Points on an axis are not in any quadrant.
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4.**S(-2.5, 0):** The y-coordinate is 0. This means the point lies directly on the **x-axis**. Points on an axis are not in any quadrant.
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Now let's evaluate the given options:
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***Point R does not lie in any quadrant:** This is **TRUE**. As we determined, R(0, -100) lies on the y-axis.
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***Points P and R lie in Quadrant III:** This is **FALSE**. P is in Quadrant IV, and R is on the y-axis.
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***Points S and Q lie in Quadrant II:** This is **FALSE**. Q is in Quadrant II, but S is on the x-axis.
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***Points R and S cannot be represented on the rectangular coordinate system:** This is **FALSE**. Any ordered pair of real numbers, including those with zero, can be precisely located on the plane.
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**Final Answer:** The only correct option is **"Point R does not lie in any quadrant"**.
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### **Question 2: Fundamentals of the Coordinate System**
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**The Question:**
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Which of the following is/are correct with respect to the rectangular coordinate system? (Multiple Select Question)
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* The horizontal line is called Y-axis
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* The point of intersection of the X and Y axes is called the origin
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* The vertical line is called X-axis
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* Any point on the coordinate plane can be represented as an ordered pair (x, y)
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**Core Concepts: Definitions**
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***x-axis:** The horizontal number line that passes through the origin.
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***y-axis:** The vertical number line that passes through the origin.
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***Origin:** The specific point $(0, 0)$ where the x-axis and y-axis intersect.
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***Ordered Pair:** The standard notation $(x, y)$ that gives the unique "address" of any point by specifying its horizontal distance (x) and vertical distance (y) from the origin.
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**Detailed Solution:**
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Let's check each statement against the definitions:
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1.**The horizontal line is called Y-axis:** This is **FALSE**. The horizontal line is the x-axis.
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2.**The point of intersection of the X and Y axes is called the origin:** This is **TRUE**. This is the definition of the origin.
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3.**The vertical line is called X-axis:** This is **FALSE**. The vertical line is the y-axis.
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4.**Any point on the coordinate plane can be represented as an ordered pair (x, y):** This is **TRUE**. This is the fundamental purpose of the coordinate system.
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**Final Answer:** The two correct statements are **"The point of intersection of the X and Y axes is called the origin"** and **"Any point on the coordinate plane can be represented as an ordered pair (x, y)"**.
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