Answer:
Explanation:
To find the total number of parts that will split line segment CD, we first need to find the distance between points C and D using the distance formula:
d(C,D) = sqrt((4 - (-6))^2 + (5 - 0)^2) = sqrt(10^2 + 5^2) = sqrt(125) = 5sqrt(5)
Next, we can use the part-to-part ratio of 3:2 to find the length of the two segments that the point P partitions CD into. Let x be the length of the segment CP and y be the length of the segment PD. Then:
x/y = 3/2
We can also use the distance formula to find the lengths of CP and PD:
d(C,P) = sqrt((0 - (-6))^2 + (3 - 0)^2) = sqrt(6^2 + 3^2) = sqrt(45) = 3sqrt(5)
d(P,D) = sqrt((4 - 0)^2 + (5 - 3)^2) = sqrt(4^2 + 2^2) = sqrt(20) = 2sqrt(5)
Since x:y = 3:2, we can write:
x = (3/5)(x + y)
y = (2/5)(x + y)
Solving this system of equations, we get:
x = 9sqrt(5)/4
y = 3sqrt(5)/4
The total number of parts that CD is split into is:
x/d(C,D) + y/d(C,D) = (9sqrt(5)/4)/(5sqrt(5)) + (3sqrt(5)/4)/(5sqrt(5)) = 12/20 + 6/20 = 18/20 = 9/10
Therefore, the line segment CD is split into 10 parts, and point P partitions it into 3 parts and 7 parts.