The point on the x- axis nearest to the point (-4,-5) is

The point on the x-axis nearest to the point \((-4, -5)\) can be found as follows:

Since the nearest point must lie on the x-axis, its y-coordinate must be zero. The general form of a point on the x-axis is \((x, 0)\). 

Now, to minimize the distance between the point \((-4, -5)\) and any point on the x-axis, we calculate the distance between \((-4, -5)\) and \((x, 0)\) using the distance formula:

\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
Substituting \((-4, -5)\) and \((x, 0)\) into the formula, we get:

\[
d = \sqrt{(x - (-4))^2 + (0 - (-5))^2} = \sqrt{(x + 4)^2 + 5^2} = \sqrt{(x + 4)^2 + 25}
\]

To minimize this distance, we observe that the x-coordinate of the nearest point must be the same as the x-coordinate of \((-4, -5)\), which is \(x = -4\). So, the nearest point on the x-axis is:

\[
(-4, 0)
\]

Thus, the point on the x-axis nearest to \((-4, -5)\) is \((-4, 0)\).

Would you like more details or have any questions?

Here are 5 related questions:
1. How do you calculate the distance between two points in 3D space?
2. What is the shortest distance between a point and a line in geometry?
3. How can you find the point on the y-axis nearest to a given point?
4. What is the reflection of a point over the x-axis or y-axis?
5. How do you determine the distance from a point to a plane in 3D?

**Tip:** When finding the nearest point on an axis, remember that the coordinate on the other axis will always be zero.

Learn how to find the point on the x-axis nearest to (-4, -5) by using the distance formula. This problem focuses on coordinate geometry, calculating distances between points, and understanding the Pythagorean Theorem. Suitable for students in Grades 8-10.

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