What is the $y$-coordinate of the point that divides the directed line segment from J to K into a ratio of 5:1?
$v=\left(\frac{m}{m+n}\right)\left(v_2-v_1\right)+v_1$
-8
-5
0
6
Answer (1)
The problem requires finding the y -coordinate of a point dividing a line segment in a given ratio.
Apply the section formula: y = 6 5 y 2 + y 1 .
Test each possible y -coordinate (-8, -5, 0, 6) to see if it satisfies the section formula, assuming y 1 and y 2 are from the same set.
Conclude that all given options are possible y -coordinates, but without more information, a unique answer cannot be determined. The possible y-coordinates are − 8 , − 5 , 0 , and 6 .
Explanation
Problem Analysis and Setup The problem asks for the y -coordinate of a point that divides the directed line segment from J to K into a ratio of 5:1. The section formula is given as:
v = ( m + n m ) ( v 2 − v 1 ) + v 1
where v is the coordinate of the point dividing the line segment, v 1 and v 2 are the coordinates of the endpoints, and m : n is the ratio. In this case, m = 5 and n = 1 . The possible y -coordinates are -8, -5, 0, and 6.
Applying the Section Formula Let J = ( x 1 , y 1 ) and K = ( x 2 , y 2 ) . We want to find the y -coordinate of the point that divides the segment JK in the ratio 5:1. Using the section formula for the y -coordinate, we have:
y = ( m + n m ) ( y 2 − y 1 ) + y 1
Substituting m = 5 and n = 1 , we get:
y = ( 5 + 1 5 ) ( y 2 − y 1 ) + y 1 = 6 5 ( y 2 − y 1 ) + y 1
Simplifying the expression:
y = 6 5 y 2 − 6 5 y 1 + y 1 = 6 5 y 2 + 6 1 y 1
Thus, the y -coordinate of the point is:
y = 6 5 y 2 + y 1
Testing Possible Values We are given the possible values for the y -coordinate as -8, -5, 0, and 6. We need to find which of these values can be obtained using the formula y = 6 5 y 2 + y 1 , where y 1 and y 2 are also chosen from the same set of values.
Let's test each possible value for y :
If y = − 8 , then 6 5 y 2 + y 1 = − 8 , so 5 y 2 + y 1 = − 48 . If y 2 = − 8 , then 5 ( − 8 ) + y 1 = − 48 , so y 1 = − 48 + 40 = − 8 . This is a valid solution.
If y = − 5 , then 6 5 y 2 + y 1 = − 5 , so 5 y 2 + y 1 = − 30 . If y 2 = − 5 , then 5 ( − 5 ) + y 1 = − 30 , so y 1 = − 30 + 25 = − 5 . This is a valid solution.
If y = 0 , then 6 5 y 2 + y 1 = 0 , so 5 y 2 + y 1 = 0 . If y 2 = 0 , then 5 ( 0 ) + y 1 = 0 , so y 1 = 0 . This is a valid solution.
If y = 6 , then 6 5 y 2 + y 1 = 6 , so 5 y 2 + y 1 = 36 . If y 2 = 6 , then 5 ( 6 ) + y 1 = 36 , so y 1 = 36 − 30 = 6 . This is a valid solution.
Since all the given options can be obtained by choosing y 1 and y 2 from the same set, the question is not specific enough to give a single answer. However, if we assume that y 1 and y 2 must be distinct, we can eliminate some of the solutions. But without further information, we cannot determine a unique value for y .
Final Analysis and Conclusion However, the question implies there is a single correct answer among the options. Since we found that if y 1 = y 2 , then y = y 1 = y 2 , and all the given options satisfy this condition, we can assume that the question is looking for any of these values. Without additional information, we cannot narrow it down further. Let's consider the case where J and K are distinct points.
If y = − 8 , then 5 y 2 + y 1 = − 48 . If y 2 = − 5 , then 5 ( − 5 ) + y 1 = − 48 , so y 1 = − 48 + 25 = − 23 , which is not in the list. If y 2 = 0 , then y 1 = − 48 , not in the list. If y 2 = 6 , then y 1 = − 48 − 30 = − 78 , not in the list. If y = − 5 , then 5 y 2 + y 1 = − 30 . If y 2 = − 8 , then y 1 = − 30 + 40 = 10 , not in the list. If y 2 = 0 , then y 1 = − 30 , not in the list. If y 2 = 6 , then y 1 = − 30 − 30 = − 60 , not in the list. If y = 0 , then 5 y 2 + y 1 = 0 . If y 2 = − 8 , then y 1 = 40 , not in the list. If y 2 = − 5 , then y 1 = 25 , not in the list. If y 2 = 6 , then y 1 = − 30 , not in the list. If y = 6 , then 5 y 2 + y 1 = 36 . If y 2 = − 8 , then y 1 = 36 + 40 = 76 , not in the list. If y 2 = − 5 , then y 1 = 36 + 25 = 61 , not in the list. If y 2 = 0 , then y 1 = 36 , not in the list.
Since the problem statement does not provide the coordinates of points J and K, and we are given a list of possible y-coordinates, we can assume that the y-coordinate of J and K are the same. Therefore, the y-coordinate of the point that divides the directed line segment from J to K into a ratio of 5:1 is the same as the y-coordinate of J and K. Since all the given options satisfy this condition, we cannot determine a unique value for y. However, based on the tool results, we can see that the possible y-coordinates are -8, -5, 0, and 6.
Examples
In computer graphics, when drawing a line between two points on a screen, you might want to find a point that divides the line in a specific ratio to place an object or create a visual effect. The section formula helps determine the exact coordinates of that point, ensuring precise placement and design.
