A line segment has endpoints A=(1,3) and B=(13,19).
a) Find the equation of the circle for which the segment AB is a diameter.
Ans: (x - 7)^2 + (y - 11)^2 = 100
b) Accurate to the nearest tenth, the circumference of this circle is _______ ?
I was able to find a)the equation but I don't know how to find the circumference, PLEASE HELP!!
You got the hardest part,
from your equation isn't the radius equal to 10
And isn't the perimeter of a circle 2pi(r) ?
Right, I was using the radius as 100, but got it now. Thanks!!
To find the circumference of a circle, you need to know either its radius or its diameter. In this case, we can use the fact that segment AB is the diameter of the circle.
a) Equation of the circle:
To find the equation of the circle with a diameter AB, we need to find the center point (h, k) of the circle. The coordinates of the center can be found by finding the midpoint of the line segment AB, which is the average of the x-coordinates and the average of the y-coordinates. So, the coordinates of the center are:
h = (x1 + x2) / 2 = (1 + 13) / 2 = 7
k = (y1 + y2) / 2 = (3 + 19) / 2 = 11
The radius of the circle can be found by taking half the length of the line segment AB, which is the distance between points A and B. The distance formula between two points (x1, y1) and (x2, y2) is:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Using this formula, the length of AB is:
d = sqrt((13 - 1)^2 + (19 - 3)^2) = sqrt(12^2 + 16^2) = sqrt(144 + 256) = sqrt(400) = 20
So, the radius of the circle is half of the length of the line segment AB, which is 20 / 2 = 10.
Therefore, the equation of the circle is:
(x - 7)^2 + (y - 11)^2 = 10^2
(x - 7)^2 + (y - 11)^2 = 100
b) Circumference of the circle:
To find the circumference of the circle, we can use the formula:
C = 2 * π * r
where C is the circumference and r is the radius.
Using the equation above, with the radius as 10, we can calculate the circumference:
C = 2 * π * 10 ≈ 20 * 3.14 ≈ 62.8
Therefore, accurate to the nearest tenth, the circumference of the circle is approximately 62.8.