To find the radius of the circle passing through points [tex]\( M \)[/tex], [tex]\( B \)[/tex], and [tex]\( C \)[/tex] where [tex]\( MBC \)[/tex] forms a right triangle with [tex]\( BC \)[/tex] as the hypotenuse, follow these steps:
1. Identify the Coordinates:
- Point [tex]\( B \)[/tex] has coordinates [tex]\((2, 6)\)[/tex].
- Point [tex]\( C \)[/tex] has coordinates [tex]\((4, 8)\)[/tex].
2. Calculate the Distance Between [tex]\( B \)[/tex] and [tex]\( C \)[/tex]:
- The formula to find the Euclidean distance between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[
\text{distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
- Applying the coordinates of [tex]\( B \)[/tex] and [tex]\( C \)[/tex]:
[tex]\[
\text{distance}_{BC} = \sqrt{(4 - 2)^2 + (8 - 6)^2}
\][/tex]
- Simplify within the square root:
[tex]\[
\text{distance}_{BC} = \sqrt{2^2 + 2^2} = \sqrt{4 + 4} = \sqrt{8} \approx 2.828
\][/tex]
3. Identify the Diameter of the Circle:
- Because [tex]\( BC \)[/tex] is the hypotenuse of the right triangle [tex]\( MBC \)[/tex], it represents the diameter of the circle according to the property of the circumcircle of a right triangle.
- Thus, the diameter [tex]\( D \)[/tex] is given by:
[tex]\[
D = \text{distance}_{BC} = \sqrt{8} \approx 2.828
\][/tex]
4. Calculate the Radius of the Circle:
- The radius [tex]\( R \)[/tex] of the circle is half of the diameter:
[tex]\[
R = \frac{D}{2} = \frac{\sqrt{8}}{2} = \sqrt{2}
\][/tex]
5. Determine the Correct Answer:
- Given the options:
A. [tex]\( 2\sqrt{2} \)[/tex]
B. [tex]\( \sqrt{2} \)[/tex]
C. [tex]\( 2 \)[/tex]
D. [tex]\( 4 \)[/tex]
- The radius calculated is [tex]\( \sqrt{2} \)[/tex], which corresponds to option B.
Therefore, the radius of the circle passing through points [tex]\( M \)[/tex], [tex]\( B \)[/tex], and [tex]\( C \)[/tex] is [tex]\( \sqrt{2} \)[/tex].
[tex]\[
\boxed{\sqrt{2}}
\][/tex]
