Final answer:
The length of segment CD is 40 units. The midpoint of segment CD is (0, 10). The point that divides the segment into a 7:3 ratio is (14, 7) and the point that divides the segment into a 2:5 ratio is (10, 10).
Step-by-step explanation:
To find the length of the line segment CD, we can use the distance formula. The distance formula is given by:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
In this case, the coordinates of point C are (-20, 10) and the coordinates of point D are (20, 10). Substituting these values into the distance formula, we get:
d = sqrt((20 - (-20))^2 + (10 - 10)^2)
d = sqrt(40^2 + 0^2)
d = sqrt(1600 + 0)
d = sqrt(1600)
d = 40
Therefore, the length of the segment CD is 40 units.
The midpoint of a line segment can be found by taking the average of the x-coordinates and the y-coordinates of the endpoints. In this case, the x-coordinate of point C is -20, the x-coordinate of point D is 20, the y-coordinate of both points is 10. So the midpoint has an x-coordinate of (-20 + 20)/2 = 0 and a y-coordinate of 10. Therefore, the midpoint of segment CD is (0, 10).
To find the point that divides the segment CD into a 7:3 ratio, we can use the section formula. The section formula is given by:
P = ((m*x2 + n*x1)/(m+n), (m*y2 + n*y1)/(m+n))
In this case, m = 7, n = 3, x1 = -20, y1 = 10, x2 = 20, y2 = 10. Substituting these values into the section formula, we get:
P = ((7*20 + 3*(-20))/(7+3), (7*10 + 3*10)/(7+3))
P = (140/10, 70/10)
P = (14, 7)
Therefore, the point that divides the segment CD into a 7:3 ratio is (14, 7).
To find the point that divides the segment CD into a 2:5 ratio, we can again use the section formula. In this case, m = 2, n = 5, x1 = -20, y1 = 10, x2 = 20, y2 = 10. Substituting these values into the section formula, we get:
P = ((2*20 + 5*(-20))/(2+5), (2*10 + 5*10)/(2+5))
P = (10, 10)
Therefore, the point that divides the segment CD into a 2:5 ratio is (10, 10).