At Westonci.ca, we provide clear, reliable answers to all your questions. Join our vibrant community and get the solutions you need. Explore our Q&A platform to find in-depth answers from a wide range of experts in different fields. Our platform offers a seamless experience for finding reliable answers from a network of knowledgeable professionals.
Sagot :
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]
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]
Thank you for trusting us with your questions. We're here to help you find accurate answers quickly and efficiently. We appreciate your time. Please revisit us for more reliable answers to any questions you may have. Find reliable answers at Westonci.ca. Visit us again for the latest updates and expert advice.