Welcome to Westonci.ca, where finding answers to your questions is made simple by our community of experts. Get accurate and detailed answers to your questions from a dedicated community of experts on our Q&A platform. Explore comprehensive solutions to your questions from a wide range of professionals on our user-friendly platform.

Which point is on the circle centered at the origin with a radius of 5 units?

Distance formula: [tex]\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}[/tex]

A. [tex]\((2, \sqrt{21})\)[/tex]

B. [tex]\((2, \sqrt{23})\)[/tex]

C. [tex]\((2, 1)\)[/tex]

D. [tex]\((2, 3)\)[/tex]


Sagot :

To determine which point is on the circle centered at the origin (0, 0) with a radius of 5 units, we need to calculate the distance of each point from the origin using the distance formula and then compare the result with the circle's radius.

The distance formula between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[ \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]

Here, our circle is centered at the origin [tex]\((0, 0)\)[/tex] and has a radius of 5 units, so we will calculate the distance of each point from the origin.

1. Point [tex]\((2, \sqrt{21})\)[/tex]:
[tex]\[ \sqrt{(2 - 0)^2 + (\sqrt{21} - 0)^2} = \sqrt{2^2 + (\sqrt{21})^2} = \sqrt{4 + 21} = \sqrt{25} = 5 \][/tex]
The distance is 5, so this point lies on the circle.

2. Point [tex]\((2, \sqrt{23})\)[/tex]:
[tex]\[ \sqrt{(2 - 0)^2 + (\sqrt{23} - 0)^2} = \sqrt{2^2 + (\sqrt{23})^2} = \sqrt{4 + 23} = \sqrt{27} \approx 5.196 \][/tex]
The distance is approximately 5.196, which is greater than 5, so this point does not lie on the circle.

3. Point [tex]\((2, 1)\)[/tex]:
[tex]\[ \sqrt{(2 - 0)^2 + (1 - 0)^2} = \sqrt{2^2 + 1^2} = \sqrt{4 + 1} = \sqrt{5} \approx 2.236 \][/tex]
The distance is approximately 2.236, which is less than 5, so this point does not lie on the circle.

4. Point [tex]\((2, 3)\)[/tex]:
[tex]\[ \sqrt{(2 - 0)^2 + (3 - 0)^2} = \sqrt{2^2 + 3^2} = \sqrt{4 + 9} = \sqrt{13} \approx 3.606 \][/tex]
The distance is approximately 3.606, which is less than 5, so this point does not lie on the circle.

Based on the calculations, the point that lies on the circle centered at the origin with a radius of 5 units is:
[tex]\[ (2, \sqrt{21}) \][/tex]
Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. We hope our answers were useful. Return anytime for more information and answers to any other questions you have. Thank you for trusting Westonci.ca. Don't forget to revisit us for more accurate and insightful answers.