If $j$ And $k$ Are Nonzero Integers, Which Pair Of Points Must Lie In The Same Quadrant?A. $(j, J$\] And $(k, K$\]B. $(i, K$\] And $(ik, Jk$\]C. $(j+k, 3$\] And $(3, J+k$\]D. $(3j,

Understanding Quadrants and Coordinate Planes

In a standard Cartesian coordinate system, the plane is divided into four quadrants based on the signs of the x and y coordinates. The quadrants are labeled as I, II, III, and IV, with quadrant I being the top-right quadrant, where both x and y coordinates are positive. Quadrant II is the top-left quadrant, where the x-coordinate is negative and the y-coordinate is positive. Quadrant III is the bottom-left quadrant, where both x and y coordinates are negative. Finally, quadrant IV is the bottom-right quadrant, where the x-coordinate is positive and the y-coordinate is negative.

Analyzing the Options

To determine which pair of points must lie in the same quadrant, we need to examine each option carefully.

Option A: $(j, j)$ and $(k, k)$

In this option, both points have the same x and y coordinates. Since $j$ and $k$ are nonzero integers, both points will have the same sign for both x and y coordinates. Therefore, both points will lie in the same quadrant.

Option B: $(i, k)$ and $(ik, jk)$

In this option, the x and y coordinates of the two points are different. The x-coordinate of the second point is $ik$, and the y-coordinate is $jk$. Since $i$, $j$, and $k$ are nonzero integers, the signs of $ik$ and $jk$ will depend on the signs of $i$, $j$, and $k$. Therefore, it is not guaranteed that both points will lie in the same quadrant.

Option C: $(j+k, 3)$ and $(3, j+k)$

In this option, the x and y coordinates of the two points are the same, but the order is reversed. Since $j$ and $k$ are nonzero integers, the sum $j+k$ will also be a nonzero integer. Therefore, both points will have the same sign for both x and y coordinates, and they will lie in the same quadrant.

Option D: $(3j, k)$ and $(k, 3j)$

In this option, the x and y coordinates of the two points are the same, but the order is reversed. Since $j$ and $k$ are nonzero integers, the product $3j$ will also be a nonzero integer. Therefore, both points will have the same sign for both x and y coordinates, and they will lie in the same quadrant.

Conclusion

Based on the analysis of each option, we can conclude that options A, C, and D are correct. In each of these options, the pair of points has the same sign for both x and y coordinates, which means they will lie in the same quadrant.

However, since the question asks for a single pair of points that must lie in the same quadrant, we need to choose one of the correct options. Based on the analysis, we can see that options A and C are the most straightforward choices.

Final Answer

The final answer is option A: $(j, j)$ and $(k, k)$.

Understanding Quadrants and Coordinate Planes

In a standard Cartesian coordinate system, the plane is divided into four quadrants based on the signs of the x and y coordinates. The quadrants are labeled as I, II, III, and IV, with quadrant I being the top-right quadrant, where both x and y coordinates are positive. Quadrant II is the top-left quadrant, where the x-coordinate is negative and the y-coordinate is positive. Quadrant III is the bottom-left quadrant, where both x and y coordinates are negative. Finally, quadrant IV is the bottom-right quadrant, where the x-coordinate is positive and the y-coordinate is negative.

Q&A

Q: What are the possible signs of the x and y coordinates in a Cartesian coordinate system?

A: The possible signs of the x and y coordinates are:

• Positive (+): The coordinate is greater than zero.
• Negative (-): The coordinate is less than zero.
• Zero (0): The coordinate is equal to zero.

Q: How are the quadrants labeled in a Cartesian coordinate system?

A: The quadrants are labeled as follows:

• Quadrant I: Top-right quadrant, where both x and y coordinates are positive.
• Quadrant II: Top-left quadrant, where the x-coordinate is negative and the y-coordinate is positive.
• Quadrant III: Bottom-left quadrant, where both x and y coordinates are negative.
• Quadrant IV: Bottom-right quadrant, where the x-coordinate is positive and the y-coordinate is negative.

Q: What is the significance of the signs of the x and y coordinates in determining the quadrant?

A: The signs of the x and y coordinates determine the quadrant in which a point lies. If both x and y coordinates are positive, the point lies in quadrant I. If the x-coordinate is negative and the y-coordinate is positive, the point lies in quadrant II. If both x and y coordinates are negative, the point lies in quadrant III. If the x-coordinate is positive and the y-coordinate is negative, the point lies in quadrant IV.

Q: How do we determine which pair of points must lie in the same quadrant?

A: To determine which pair of points must lie in the same quadrant, we need to examine the signs of the x and y coordinates of each point. If both points have the same sign for both x and y coordinates, they will lie in the same quadrant.

Q: What are the possible pairs of points that must lie in the same quadrant?

A: Based on the analysis, the possible pairs of points that must lie in the same quadrant are:

• $(j, j)$ and $(k, k)$
• $(j+k, 3)$ and $(3, j+k)$
• $(3j, k)$ and $(k, 3j)$

Q: Why are these pairs of points guaranteed to lie in the same quadrant?

A: These pairs of points are guaranteed to lie in the same quadrant because they have the same sign for both x and y coordinates. In the case of $(j, j)$ and $(k, k)$, both points have the same sign for both x and y coordinates. In the case of $(j+k, 3)$ and $(3, j+k)$, both points have the same sign for both x and y coordinates. In the case of $(3j, k)$ and $(k, 3j)$, both points have the same sign for both x and y coordinates.

Conclusion

In conclusion, the pair of points that must lie in the same quadrant is $(j, j)$ and $(k, k)$. This is because both points have the same sign for both x and y coordinates, which means they will lie in the same quadrant.

Final Answer

The final answer is option A: $(j, j)$ and $(k, k)$.