The Perpendicular Bisector

Oct 13, 2025 by talynt.org 129 views

Understanding the Problem

To find the point P on the x-axis equidistant from points A(-1,0) and B(5,0), we need to understand the concept of distance and midpoint. The distance between two points in a coordinate plane is given by the formula:

${ \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} }$

However, since point P lies on the x-axis, its y-coordinate is 0. Therefore, we can simplify the distance formula to:

${ \sqrt{(x_2 - x_1)^2} }$

Finding the Midpoint

Since point P is equidistant from points A and B, it must lie on the perpendicular bisector of the line segment AB. The midpoint of the line segment AB can be found using the formula:

${ \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) }$

Substituting the coordinates of points A and B, we get:

${ \left( \frac{-1 + 5}{2}, \frac{0 + 0}{2} \right) = \left( \frac{4}{2}, 0 \right) = (2, 0) }$

The Perpendicular Bisector

The perpendicular bisector of the line segment AB passes through the midpoint (2, 0). Since point P lies on the x-axis, it must also lie on the perpendicular bisector. Therefore, the x-coordinate of point P is also 2.

Conclusion

Based on the above analysis, the point P on the x-axis equidistant from points A(-1,0) and B(5,0) is (2, 0).

Answer

The correct answer is (a) (2.0).

Explanation

The point P on the x-axis equidistant from points A and B is the midpoint of the line segment AB, which is (2, 0).

Understanding the Problem

${ \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} }$

However, since point P lies on the x-axis, its y-coordinate is 0. Therefore, we can simplify the distance formula to:

${ \sqrt{(x_2 - x_1)^2} }$

Q&A

Q: What is the distance formula?

A: The distance formula is given by:

${ \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} }$

Q: Why is the distance formula simplified for point P?

A: Since point P lies on the x-axis, its y-coordinate is 0. Therefore, the distance formula can be simplified to:

${ \sqrt{(x_2 - x_1)^2} }$

Q: How is the midpoint of the line segment AB found?

A: The midpoint of the line segment AB can be found using the formula:

${ \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) }$

Q: What is the x-coordinate of the midpoint of the line segment AB?

A: The x-coordinate of the midpoint of the line segment AB is:

${ \frac{-1 + 5}{2} = \frac{4}{2} = 2 }$

Q: Why is the x-coordinate of point P also 2?

A: Since point P lies on the x-axis and is equidistant from points A and B, it must lie on the perpendicular bisector of the line segment AB. The perpendicular bisector passes through the midpoint (2, 0), so the x-coordinate of point P is also 2.

Q: What is the correct answer?

A: The correct answer is (a) (2.0).

Conclusion

In this article, we have discussed the problem of finding the point P on the x-axis equidistant from points A(-1,0) and B(5,0). We have used the concept of distance and midpoint to find the solution. The x-coordinate of point P is 2, and the correct answer is (a) (2.0).

Frequently Asked Questions

Q: What is the distance formula?

A: The distance formula is given by:

${ \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} }$

Q: How is the midpoint of the line segment AB found?

A: The midpoint of the line segment AB can be found using the formula:

${ \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) }$

Q: What is the x-coordinate of the midpoint of the line segment AB?

A: The x-coordinate of the midpoint of the line segment AB is:

${ \frac{-1 + 5}{2} = \frac{4}{2} = 2 }$