Answer: 12x - 20 units
Explanation:
To find the length of segment CD, you can use the formula for the distance between two points, which is given by the midpoint formula:
Midpoint formula: M = (1/2)(C + D)
In this case, C and D are the endpoints of segment CD, and M is the midpoint. The formula for the distance between C and D is:
CD = 2 * MD
Now, let's use the given information:
CM = 10(x - 3) units
MD = 2(3x - 5) units
First, find the coordinates of M using the midpoint formula:
M = (1/2)(C + D)
Since M is the midpoint, it is also the average of C and D:
M = (C + D)/2
Now, plug in the expressions for CM and MD:
M = [(10(x - 3), 2(3x - 5))/2
Simplify the expression inside the brackets:
M = (5(x - 3), 3x - 5)
Now, let's find the coordinates of C and D:
C = (10(x - 3), 0)
D = (0, 2(3x - 5))
Now, apply the midpoint formula to find M:
M = [(C + D)/2]
M = [(10(x - 3), 0 + 0, 2(3x - 5))/2]
Now, simplify further:
M = [(10(x - 3), 2(3x - 5))/2]
Now, we have the coordinates of M, which are (5(x - 3), 3x - 5).
To find CD, use the distance formula:
CD = 2 * MD
CD = 2 * [2(3x - 5)]
CD = 4(3x - 5)
Now, simplify:
CD = 12x - 20
So, the length of segment CD is 12x - 20 units.