1. In the xy-coordinate plane, point A is the midpoint of the segment with endpoints (2,4) and (-4,-4). What is the distance from point A to the origin?
I know how to find the distance between the two points- 10. But I don't know where to go from there.
2. The radius of circle B is 125 percent of the radius of circle A, and the radius of circle C is 80 percent of the radius of circle B. If the area of circle A is 50pi, what is the area of circle C?
Thanks in advance!
well, you need to find the midpoint. That is at the average of the ends: (-1,0). Now just find the distance from there to (0,0).
If the radii are a,b,c we have
b = 5/4 a
c = 4/5 b = 4/5 * 5/4 a = a
so, the radii are the same, and hence the areas as well.
Note that we don't even have to remember that the areas scale as the square of the radii.
How do you find the "average of the ends"? What does that mean and how do you get -1?
1. To find the distance from point A to the origin, you can use the distance formula. The distance formula is sqrt((x2 - x1)^2 + (y2 - y1)^2), where (x1, y1) represents the coordinates of the origin and (x2, y2) represents the coordinates of point A.
In this case, the origin is (0, 0), and point A is the midpoint of the segment with endpoints (2, 4) and (-4, -4). To find the coordinates of point A, you can take the average of the x-coordinates and the average of the y-coordinates of the two endpoints.
The x-coordinate of point A: (2 + (-4)) / 2 = -2/2 = -1
The y-coordinate of point A: (4 + (-4)) / 2 = 0/2 = 0
So point A has coordinates (-1, 0). Now, you can substitute the values into the distance formula:
Distance from A to the origin = sqrt((-1 - 0)^2 + (0 - 0)^2)
= sqrt((-1)^2 + 0^2)
= sqrt(1 + 0)
= sqrt(1)
= 1
Therefore, the distance from point A to the origin is 1.
2. The problem states that the radius of circle B is 125 percent of the radius of circle A. This means that the radius of circle B is 1.25 times the radius of circle A. Similarly, the radius of circle C is 80 percent of the radius of circle B, which means the radius of circle C is 0.8 times the radius of circle B.
Given that the area of circle A is 50π, we can use the formula for the area of a circle, which is A = πr^2. In this case, since we know the area of circle A and want to find the area of circle C, we need to find the radius of circle A first.
50π = πr^2
Dividing both sides by π: 50 = r^2
Taking the square root of both sides: r = √50
Now, we can calculate the radius of circle B:
Radius of circle B = 1.25 * √50
Finally, we can calculate the area of circle C using the known radius of circle B:
Area of circle C = π * (0.8 * 1.25 * √50)^2
Simplifying the expression within the parentheses:
Area of circle C = π * (1 * √50)^2
Simplifying further:
Area of circle C = π * 50
Therefore, the area of circle C is 50π.