A scientist is plotting the circular path of a particle on a coordinate plane for a lab experiment. The scientist knows the path is a perfect circle and that the particle starts at ordered pair (-5, -11). When the particle is halfway around the circle, the particle is at ordered pair (11,19). The segment formed by connecting these two points has the center of the circle at it’s midpoint. A.)What is the ordered pair that represents the center of the circle? B.) What is the length of the radius, in units, of the circle? C.) Explain why the particle can never pass through a point with an X-coordinate of 24 as long as it stays on the circular path.
As the question suggests, the line joining your two points is a diameter.
So the centre is the midpoint, which is
((-5+11)/2 , (-11+19)/2 )
= (3, 4)
radius = √(3+5)^2 + (4+11)^2
= √(64+225) = √289 = 17
equation of circle:
(x-3)^2 + (y-4)^2 = 289
if x = 24
21^2 + (y-4)^2 = 289
(y-4)^2 = -152
to solve this , we would take √ of both sides, but you can't take the √ of a negative.
So there is point which has x = 24 as its x coordinate.
A.) The ordered pair that represents the center of the circle can be found by taking the average of the x-coordinates and the y-coordinates of the starting point and the halfway point. So, the x-coordinate of the center would be (-5 + 11) / 2 = 3, and the y-coordinate would be (-11 + 19) / 2 = 4. Therefore, the ordered pair representing the center of the circle is (3, 4).
B.) The length of the radius can be found by calculating the distance between the center of the circle and either the starting point or the halfway point. Using the distance formula, we can calculate the distance between (3, 4) and (-5, -11), which gives us the radius. The distance can be calculated using the formula √[(x2 - x1)^2 + (y2 - y1)^2]. So, the radius is √[(3 - (-5))^2 + (4 - (-11))^2] = √[(8)^2 + (15)^2] = √(64 + 225) = √289 = 17. Therefore, the radius of the circle is 17 units.
C.) The particle can never pass through a point with an X-coordinate of 24 because the center of the circle is located at (3, 4). Since the radius is 17 units, it means that the distance from the center of the circle to any point on the circle will always be 17 units. So, if the particle stays on the circular path, it can never reach a point with an X-coordinate greater than (3 + 17) = 20 or smaller than (3 - 17) = -14. Therefore, it can never pass through a point with an X-coordinate of 24.
To find the ordered pair that represents the center of the circle, we can use the midpoint formula. The midpoint of the line segment connecting (-5, -11) and (11, 19) will give us the center of the circle.
A.) Finding the ordered pair representing the center of the circle:
Let's use the midpoint formula:
Midpoint = ((x1 + x2)/2, (y1 + y2)/2)
Substituting the values, we have:
Midpoint = ((-5 + 11)/2, (-11 + 19)/2)
Midpoint = (6/2, 8/2)
Midpoint = (3, 4)
So, the ordered pair representing the center of the circle is (3, 4).
B.) Finding the length of the radius of the circle:
The radius of the circle is the distance from the center to any point on the circle. We need to find the distance between the center (3, 4) and one of the points on the circle, let's say (-5, -11).
Using the distance formula, the length of the radius can be found as follows:
Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Substituting the values, we have:
Distance = sqrt((-5 - 3)^2 + (-11 - 4)^2)
Distance = sqrt((-8)^2 + (-15)^2)
Distance = sqrt(64 + 225)
Distance = sqrt(289)
Distance = 17
So, the length of the radius of the circle is 17 units.
C.) The particle can never pass through a point with an X-coordinate of 24 as long as it stays on the circular path because the x-coordinate of the center of the circle is 3. The x-coordinate of any point on the circle must be within the range (-17, 23) since the radius is 17 units. As 24 is outside this range, the particle cannot pass through a point with an x-coordinate of 24.
A.) To find the center of the circle, we can use the midpoint formula. The midpoint formula states that the coordinates of the midpoint of a line segment with endpoints (x1, y1) and (x2, y2) are given by:
Midpoint = ((x1 + x2)/2, (y1 + y2)/2)
In this case, the endpoint coordinates are (-5, -11) and (11, 19), and we want to find the midpoint. Plugging the values into the midpoint formula, we get:
Midpoint = ((-5 + 11)/2, (-11 + 19)/2)
= (6/2, 8/2)
= (3, 4)
Therefore, the ordered pair (3, 4) represents the center of the circle.
B.) To find the length of the radius of the circle, we can use the distance formula. The distance formula states that the distance between two points (x1, y1) and (x2, y2) is given by:
Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
In this case, the two points are (-5, -11) and (3, 4). Plugging the values into the distance formula, we get:
Distance = sqrt((3 - (-5))^2 + (4 - (-11))^2)
= sqrt((8)^2 + (15)^2)
= sqrt(64 + 225)
= sqrt(289)
= 17
Therefore, the length of the radius of the circle is 17 units.
C.) The particle can never pass through a point with an x-coordinate of 24 because the given information states that the particle starts at (-5, -11), moves halfway around the circle, and ends up at (11, 19). The x-coordinate of the particle changes from -5 to 11, which is a total change of 16 units. This means the x-coordinate can only go up to 11 + 16 = 27 or down to -5 - 16 = -21. It cannot reach the x-coordinate of 24, as it is outside the range of possible values for the particle's position on the circular path. Therefore, the particle can never pass through a point with an x-coordinate of 24 as long as it stays on the circular path.