A circle of area 25 pi is graphed in the coordinate plane. If its center has coordinates (8,0), which of the following points MUST lie on the circumference of the circle?
a. (0,0)
b. (0,5)
c. (16,0)
d. (3,0)
e. (0,8)
please answer and explain
radius is 5, so since (3,0) is 5 units to the left of (8,0), that's our point.
(d)
To determine which of the following points must lie on the circumference of the circle, we need to calculate the radius of the circle first.
The area of a circle is given by the formula: A = πr^2, where A is the area and r is the radius.
In this case, the area of the circle is given as 25π. Therefore, we can derive the following equation:
25π = πr^2
To solve for the radius, we divide both sides of the equation by π:
25 = r^2
Taking the square root of both sides, we find:
r = √(25) = 5
Since we now know that the radius of the circle is 5, we can check which of the given points are at a distance of 5 units from the center (8,0).
a. (0,0):
The distance between (8,0) and (0,0) can be calculated using the distance formula: d = √((x2 - x1)^2 + (y2 - y1)^2)
d = √((0 - 8)^2 + (0 - 0)^2)
= √((0 - 8)^2 + 0^2)
= √((-8)^2 + 0)
= √(64)
= 8
As the distance is 8 units, point (0,0) is not on the circumference of the circle.
b. (0,5):
The distance between (8,0) and (0,5) can be calculated using the distance formula:
d = √((0 - 8)^2 + (5 - 0)^2)
= √((-8)^2 + 5^2)
= √(64 + 25)
= √(89)
The distance is √(89), which is not equal to 5. Therefore, point (0,5) is not on the circumference of the circle.
c. (16,0):
The distance between (8,0) and (16,0) is simply the difference between the x-coordinates, which is 16 - 8 = 8 units. Point (16,0) is not on the circumference.
d. (3,0):
The distance between (8,0) and (3,0) is simply the difference between the x-coordinates, which is 8 - 3 = 5 units. Point (3,0) is on the circumference of the circle.
e. (0,8):
The distance between (8,0) and (0,8) can be calculated using the distance formula:
d = √((0 - 8)^2 + (8 - 0)^2)
= √((-8)^2 + 8^2)
= √(64 + 64)
= √(128)
The distance is √(128), which is not equal to 5. Therefore, point (0,8) is not on the circumference of the circle.
In conclusion, the only point that must lie on the circumference of the circle is option d: (3,0).