Step 1: Understanding the problem:
We are given that the diameter of a circle has a length of 6 cm, and one end of the diameter is at the point \( (-4, 0) \) on the x-axis. We need to find the coordinates of the other end of the diameter, which also lies on the x-axis.
Step 2: The midpoint of the diameter:
Since the circle's center is the midpoint of the diameter, we can find the center by averaging the x-coordinates of the two ends of the diameter.
Let the coordinates of the other end of the diameter be \( (x, 0) \), where \( x \) is the unknown value we need to find.
The midpoint of the diameter is the average of the x-coordinates of the two endpoints, i.e., \( \left( \frac{-4 + x}{2}, 0 \right) \).
Also, the center of the circle is halfway along the diameter, and the length of the diameter is 6 cm. Thus, the radius is 3 cm.
Step 3: Using the distance formula:
The distance between the point \( (-4, 0) \) and the center of the circle is the radius of the circle, which is 3 cm. We can use the distance formula to find the x-coordinate of the other end of the diameter.
The distance between \( (-4, 0) \) and the center \( \left( \frac{-4 + x}{2}, 0 \right) \) is 3 cm, so:
\[
\left| -4 - \frac{-4 + x}{2} \right| = 3
\]
Simplifying the expression inside the absolute value:
\[
\left| -4 - \frac{-4 + x}{2} \right| = \left| \frac{-8 - (-4 + x)}{2} \right| = \left| \frac{-8 + 4 - x}{2} \right| = \left| \frac{-4 - x}{2} \right| = 3
\]
Now, solve for \( x \):
\[
\left| -4 - x \right| = 6
\]
This gives two possible cases:
Case 1: \( -4 - x = 6 \)
\[
-x = 6 + 4 \quad \Rightarrow \quad x = -10
\]
Case 2: \( -4 - x = -6 \)
\[
-x = -6 + 4 \quad \Rightarrow \quad x = 2
\]
Step 4: Conclusion:
The other end of the diameter lies at \( (2, 0) \) since it is a positive x-coordinate that makes sense with the given diameter length. Hence, the coordinates of the other end of the diameter are \( (2, 0) \).