Question

A circle is drawn on the coordinate plane centered at (-2, 3) with a radius of 6 units.

Which of the following coordinates would be inside of the circle?
a.(-7, -1)
b.(3,6)
c.(-2,9)
d.(2,-2)

Answer 1

Coordinate b (3,6) is inside the circle, coordinate a (-7, -1) and d (2, -2) are outside the circle, and coordinate c is on the circumference of the circle.

Step-by-step explanation:

To determine which coordinates are inside the circle, we need to calculate the distance between each coordinate and the center of the circle. We can use the distance formula: d = sqrt((x2 - x1)^2 + (y2 - y1)^2).

For coordinate a: d = sqrt((-7 - (-2))^2 + (-1 - 3)^2) = sqrt(25 + 16) = 7.81. Since 7.81 is greater than the radius of 6 units, coordinate a is outside the circle.

For coordinate b: d = sqrt((3 - (-2))^2 + (6 - 3)^2) = sqrt(25 + 9) = 5.83. Since 5.83 is less than the radius of 6 units, coordinate b is inside the circle.

For coordinate c: d = sqrt((-2 - (-2))^2 + (9 - 3)^2) = sqrt(0 + 36) = 6. Since 6 is equal to the radius of 6 units, coordinate c is on the circumference of the circle.

For coordinate d: d = sqrt((2 - (-2))^2 + (-2 - 3)^2) = sqrt(16 + 25) = 7.81. Since 7.81 is greater than the radius of 6 units, coordinate d is outside the circle.

Answer 2

Using the distance formula, points a.(-7, -1), b.(3, 6), and d.(2, -2) are inside the circle with a center at (-2, 3) and radius of 6, while point c.(-2, 9) is on the circumference.

Step-by-step explanation:

The question asks which coordinate would be inside a circle on a coordinate plane. The circle has a center at (-2, 3) and a radius of 6 units. To determine if a point is inside the circle, we subtract the center coordinates from the point's coordinates and calculate the square root of the sum of the squared differences. If the result is less than the radius, the point is inside the circle.

For point a.(-7, -1), the calculation is:

√((-7 - (-2))² + (-1 - 3)² ) = √(25 + 16) = √41 which is less than 6. Therefore, (-7, -1) is inside the circle.

For point b.(3, 6), the calculation is:

√((3 - (-2))² + (6 - 3)² ) = √(25 + 9) = √34 which is also less than 6. Therefore, (3, 6) is inside the circle.

For point c.(-2, 9), the calculation is:

√((-2 - (-2))² + (9 - 3)² ) = √(0 + 36) = 6 which is equal to the radius, so (-2, 9) is located at the circumference of the circle and not inside it.

For point d.(2, -2), the calculation is:

√((2 - (-2))² + (-2 - 3)² ) = √(16 + 25) = √41 which is less than 6, so (2, -2) is inside the circle.

Therefore, three of the points a, b, and d are inside the circle, while point c is on the circumference.