Final answer:
The length of segment CD is 60 units, the midpoint is (0, 10), the point dividing the segment into a 7:3 ratio is (8, 10), and the point in a 2:5 ratio is (-10, 10).
Step-by-step explanation:
Length, Midpoint, and Division Points of a Line Segment
The length of the segment CD with points C (-20, 10) and D (20, 10) is found using the distance formula which is
√((x2 - x1)² + (y2 - y1)²). As the y-coordinates are the same, the length is simply the difference in x-coordinates, which is 40 - (-20) = 60 units.
The midpoint of segment CD is calculated using the midpoint formula, which is ((x1 + x2)/2, (y1 + y2)/2). Thus, the midpoint is (0, 10).
To find the point that divides the segment into a 7:3 ratio, we use the section formula. For a division in ratio m:n, the coordinates of the division point are ((mx2 + nx1)/(m + n), (my2 + ny1)/(m + n)).
Hence, the dividing point for 7:3 is ((7*20 + 3*(-20))/(7+3), (7*10 + 3*10)/(7+3)) which simplifies to (8, 10).
The point that divides the segment into a 2:5 ratio is found in the same manner.
The dividing point for 2:5 is ((2*20 + 5*(-20))/(2+5), (2*10 + 5*10)/(2+5)) which simplifies to (-10, 10).