At Westonci.ca, we connect you with the answers you need, thanks to our active and informed community. Our platform provides a seamless experience for finding reliable answers from a network of experienced professionals. Experience the ease of finding precise answers to your questions from a knowledgeable community of experts.
Sagot :
Certainly! To find the equation of the circle that has segment [tex]\( PQ \)[/tex] as its diameter, we need to follow these steps:
### 1. Find the midpoint of the segment PQ:
The midpoint of a segment is given by the average of the x-coordinates and the average of the y-coordinates of the endpoints.
Given points [tex]\( P = (-3, 5) \)[/tex] and [tex]\( Q = (1, 9) \)[/tex], the midpoint, which will be the center of the circle, is calculated as:
[tex]\[ \text{Midpoint} = \left( \frac{-3 + 1}{2}, \frac{5 + 9}{2} \right) \][/tex]
[tex]\[ \text{Midpoint} = (-1.0, 7.0) \][/tex]
### 2. Find the radius of the circle:
The radius of the circle is half the length of segment [tex]\( PQ \)[/tex]. The length of [tex]\( PQ \)[/tex] can be found using the distance formula:
[tex]\[ \text{Distance} = \sqrt{(1 - (-3))^2 + (9 - 5)^2} \][/tex]
[tex]\[ \text{Distance} = \sqrt{(4)^2 + (4)^2} \][/tex]
[tex]\[ \text{Distance} = \sqrt{16 + 16} \][/tex]
[tex]\[ \text{Distance} = \sqrt{32} \][/tex]
[tex]\[ \text{Radius} = \frac{\sqrt{32}}{2} = \sqrt{8} \][/tex]
### 3. Write the equation of the circle:
The general form of the equation of a circle with center [tex]\((h, k)\)[/tex] and radius [tex]\( r \)[/tex] is:
[tex]\[ (x - h)^2 + (y - k)^2 = r^2 \][/tex]
Given:
- The center [tex]\((h, k) = (-1, 7)\)[/tex]
- The radius squared [tex]\( r^2 = 8 \)[/tex]
The equation of the circle becomes:
[tex]\[ (x + 1)^2 + (y - 7)^2 = 8 \][/tex]
Hence, the equation of the circle that has segment [tex]\( PQ \)[/tex] as its diameter is:
[tex]\[ (x + 1)^2 + (y - 7)^2 = 8.000000000000002 \][/tex]
### 1. Find the midpoint of the segment PQ:
The midpoint of a segment is given by the average of the x-coordinates and the average of the y-coordinates of the endpoints.
Given points [tex]\( P = (-3, 5) \)[/tex] and [tex]\( Q = (1, 9) \)[/tex], the midpoint, which will be the center of the circle, is calculated as:
[tex]\[ \text{Midpoint} = \left( \frac{-3 + 1}{2}, \frac{5 + 9}{2} \right) \][/tex]
[tex]\[ \text{Midpoint} = (-1.0, 7.0) \][/tex]
### 2. Find the radius of the circle:
The radius of the circle is half the length of segment [tex]\( PQ \)[/tex]. The length of [tex]\( PQ \)[/tex] can be found using the distance formula:
[tex]\[ \text{Distance} = \sqrt{(1 - (-3))^2 + (9 - 5)^2} \][/tex]
[tex]\[ \text{Distance} = \sqrt{(4)^2 + (4)^2} \][/tex]
[tex]\[ \text{Distance} = \sqrt{16 + 16} \][/tex]
[tex]\[ \text{Distance} = \sqrt{32} \][/tex]
[tex]\[ \text{Radius} = \frac{\sqrt{32}}{2} = \sqrt{8} \][/tex]
### 3. Write the equation of the circle:
The general form of the equation of a circle with center [tex]\((h, k)\)[/tex] and radius [tex]\( r \)[/tex] is:
[tex]\[ (x - h)^2 + (y - k)^2 = r^2 \][/tex]
Given:
- The center [tex]\((h, k) = (-1, 7)\)[/tex]
- The radius squared [tex]\( r^2 = 8 \)[/tex]
The equation of the circle becomes:
[tex]\[ (x + 1)^2 + (y - 7)^2 = 8 \][/tex]
Hence, the equation of the circle that has segment [tex]\( PQ \)[/tex] as its diameter is:
[tex]\[ (x + 1)^2 + (y - 7)^2 = 8.000000000000002 \][/tex]
Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. We hope this was helpful. Please come back whenever you need more information or answers to your queries. We're glad you visited Westonci.ca. Return anytime for updated answers from our knowledgeable team.