To determine the center and radius of a circle given the endpoints of its diameter, we can follow these steps:
1. **Identify the coordinates of endpoints:** Here, we’re given two points, P(10, 2) and Q(46, y). However, from the problem, we need to also establish the y-coordinate for Q to proceed. For this explanation, let’s assume the y-coordinate for Q is also 2, making Q(46, 2).
2. **Calculate the center of the circle:** The center of the circle lies at the midpoint of the diameter. We can calculate the midpoint (center) using the formula:
C = ((x1 + x2) / 2, (y1 + y2) / 2)
For our points:
C = ((10 + 46) / 2, (2 + 2) / 2) = (56 / 2, 4 / 2) = (28, 2)
Thus, the center of the circle is at the point C(28, 2).
3. **Calculate the radius of the circle:** The radius is half the length of the diameter, which can be calculated using the distance formula to find the length of the segment PQ:
d = √((x2 - x1)² + (y2 - y1)²)
Let’s substitute our points:
d = √((46 - 10)² + (2 - 2)²) = √((36)² + (0)²) = √(1296 + 0) = √1296 = 36
Since the diameter is 36, we find the radius by dividing this length by 2:
Radius = Diameter / 2 = 36 / 2 = 18
Therefore, the radius of the circle is 18.
To summarize:
- The center of the circle is at C(28, 2).
- The radius of the circle is 18.