A circle has an area of approximately 78.5 square units. If the center of the circle is at Left-parenthesis 2 comma 4 right-parenthesis, which of the following points is on the circle?
(1 point)
Responses
Left-parenthesis negative 6 comma 4 right-parenthesis
Image with alt text: Left-parenthesis negative 6 comma 4 right-parenthesis
Left-parenthesis 2 comma negative 1 right-parenthesis
Image with alt text: Left-parenthesis 2 comma negative 1 right-parenthesis
Left-parenthesis negative 4 comma negative 4 right-parenthesis
Image with alt text: Left-parenthesis negative 4 comma negative 4 right-parenthesis
Left-parenthesis 2 comma negative 9 right-parenthesis
The point that is on the circle is (2, -1).
To determine which point is on the circle, we need to find the distance between each point and the center of the circle. If the distance is equal to the radius of the circle, then the point is on the circle.
Let's calculate the distances between each point and the center of the circle.
Distance between (−6, 4) and (2, 4):
√((-6 - 2)² + (4 - 4)²)
√((-8)² + (0)²)
√(64 + 0)
√64
8
Distance between (2, -1) and (2, 4):
√((2 - 2)² + (-1 - 4)²)
√((0)² + (-5)²)
√(0 + 25)
√25
5
Distance between (-4, -4) and (2, 4):
√((-4 - 2)² + (-4 - 4)²)
√((-6)² + (-8)²)
√(36 + 64)
√100
10
Distance between (2, -9) and (2, 4):
√((2 - 2)² + (-9 - 4)²)
√((0)² + (-13)²)
√(0 + 169)
√169
13
The distance between each point and the center of the circle is:
- (−6, 4): 8
- (2, -1): 5
- (−4, −4): 10
- (2, -9): 13
The radius of the circle is the square root of the area divided by pi. In this case, the area is approximately 78.5 square units, so the radius is approximately √(78.5/π) ≈ 4.981.
Since the only point with a distance of approximately 4.981 to the center of the circle is (−6, 4), the point (−6, 4) is on the circle.