Finding the Coordinates of P and Q

Oct 13, 2025 by miodeya.com 119 views

The Line Represented by x/4 + y/6 = 1 Intersects the X Axis and Y Axis Respectively at P and Q

Understanding the Problem

The given equation of the line is x/4 + y/6 = 1. We are asked to find the coordinates of the midpoint of the line segment PQ, where P and Q are the points at which the line intersects the x-axis and y-axis respectively.

Finding the Coordinates of P and Q

To find the coordinates of P and Q, we need to find the points at which the line intersects the x-axis and y-axis respectively.

When the line intersects the x-axis, the value of y is 0. Substituting y = 0 in the equation of the line, we get:

x/4 + 0/6 = 1

Simplifying the equation, we get:

x/4 = 1

Multiplying both sides of the equation by 4, we get:

x = 4

Therefore, the coordinates of P are (4, 0).

When the line intersects the y-axis, the value of x is 0. Substituting x = 0 in the equation of the line, we get:

0/4 + y/6 = 1

Simplifying the equation, we get:

y/6 = 1

Multiplying both sides of the equation by 6, we get:

y = 6

Therefore, the coordinates of Q are (0, 6).

Finding the Midpoint of PQ

The midpoint of a line segment is the point that divides the line segment into two equal parts. The coordinates of the midpoint of a line segment with endpoints (x1, y1) and (x2, y2) are given by:

((x1 + x2)/2, (y1 + y2)/2)

In this case, the endpoints of the line segment PQ are (4, 0) and (0, 6). Therefore, the coordinates of the midpoint of PQ are:

((4 + 0)/2, (0 + 6)/2)

Simplifying the coordinates, we get:

(2, 3)

Therefore, the coordinates of the midpoint of the line segment PQ are (2, 3).

Conclusion

In this article, we have discussed how to find the coordinates of the midpoint of the line segment PQ, where P and Q are the points at which the line intersects the x-axis and y-axis respectively. We have used the equation of the line x/4 + y/6 = 1 to find the coordinates of P and Q, and then used the midpoint formula to find the coordinates of the midpoint of PQ. The coordinates of the midpoint of PQ are (2, 3).

Key Takeaways

The equation of the line x/4 + y/6 = 1 intersects the x-axis and y-axis respectively at P and Q.
The coordinates of P are (4, 0) and the coordinates of Q are (0, 6).
The coordinates of the midpoint of PQ are (2, 3).

Final Answer

The final answer is (2, 3).
The Line Represented by x/4 + y/6 = 1 Intersects the X Axis and Y Axis Respectively at P and Q: Q&A

Understanding the Problem

Q: What is the equation of the line?

A: The equation of the line is x/4 + y/6 = 1.

Q: How do we find the coordinates of P and Q?

A: To find the coordinates of P and Q, we need to find the points at which the line intersects the x-axis and y-axis respectively. When the line intersects the x-axis, the value of y is 0. Substituting y = 0 in the equation of the line, we get x/4 = 1. Multiplying both sides of the equation by 4, we get x = 4. Therefore, the coordinates of P are (4, 0). When the line intersects the y-axis, the value of x is 0. Substituting x = 0 in the equation of the line, we get y/6 = 1. Multiplying both sides of the equation by 6, we get y = 6. Therefore, the coordinates of Q are (0, 6).

Q: How do we find the midpoint of PQ?

A: The midpoint of a line segment is the point that divides the line segment into two equal parts. The coordinates of the midpoint of a line segment with endpoints (x1, y1) and (x2, y2) are given by ((x1 + x2)/2, (y1 + y2)/2). In this case, the endpoints of the line segment PQ are (4, 0) and (0, 6). Therefore, the coordinates of the midpoint of PQ are ((4 + 0)/2, (0 + 6)/2) = (2, 3).

Q: What are the coordinates of the midpoint of PQ?

A: The coordinates of the midpoint of PQ are (2, 3).

Q: Why is the midpoint formula important?

A: The midpoint formula is important because it allows us to find the coordinates of the midpoint of a line segment given the coordinates of the endpoints. This is useful in a variety of applications, including geometry and trigonometry.

Q: Can you give an example of how to use the midpoint formula?

A: Yes, let's say we have a line segment with endpoints (2, 3) and (4, 5). To find the coordinates of the midpoint of this line segment, we can use the midpoint formula: ((2 + 4)/2, (3 + 5)/2) = (3, 4).

Q: What are some common applications of the midpoint formula?

A: The midpoint formula has a variety of applications, including:

Finding the coordinates of the midpoint of a line segment
Calculating the distance between two points
Finding the equation of a line given two points
Solving problems in geometry and trigonometry

Q: Is the midpoint formula difficult to use?

A: No, the midpoint formula is relatively simple to use. It requires only a few basic steps, including adding the x-coordinates and y-coordinates of the endpoints and dividing by 2.

Q: Can you give a real-world example of how the midpoint formula is used?

A: Yes, let's say we are architects designing a building. We need to find the midpoint of a line segment that represents the length of a wall. We can use the midpoint formula to find the coordinates of the midpoint of this line segment, which will give us the location of the wall.

Conclusion

In this article, we have discussed the equation of the line x/4 + y/6 = 1 and how to find the coordinates of the midpoint of the line segment PQ. We have also answered some common questions about the midpoint formula and its applications. The midpoint formula is a useful tool in geometry and trigonometry, and it has a variety of real-world applications.