To find the length of the diameter of the circle [tex]\( Q \)[/tex] centered at [tex]\( (3,-2) \)[/tex] and passing through the point [tex]\( R(7,1) \)[/tex], we follow these steps:
1. Identify the center and a point on the circle:
- Center of the circle [tex]\( Q \)[/tex]: [tex]\( (3, -2) \)[/tex]
- Point [tex]\( R \)[/tex] on the circle: [tex]\( (7, 1) \)[/tex]
2. Calculate the radius of the circle:
- To find the radius, we use the distance formula to calculate the distance between the center of the circle and point [tex]\( R \)[/tex]. The distance formula is:
[tex]\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
- Plug in the coordinates of the center [tex]\( (3, -2) \)[/tex] and point [tex]\( R(7, 1) \)[/tex]:
[tex]\[
d = \sqrt{(7 - 3)^2 + (1 + 2)^2}
\][/tex]
[tex]\[
d = \sqrt{(4)^2 + (3)^2}
\][/tex]
[tex]\[
d = \sqrt{16 + 9}
\][/tex]
[tex]\[
d = \sqrt{25}
\][/tex]
[tex]\[
d = 5
\][/tex]
- The radius of the circle is [tex]\( 5 \)[/tex] units.
3. Calculate the diameter of the circle:
- The diameter [tex]\( D \)[/tex] of a circle is twice the radius. So:
[tex]\[
D = 2 \times \text{radius}
\][/tex]
[tex]\[
D = 2 \times 5
\][/tex]
[tex]\[
D = 10
\][/tex]
4. Choose the correct answer:
- From the given options, the length of the diameter is [tex]\( \boxed{10} \)[/tex].
