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)
Apply the section formula to find the y -coordinate of the point dividing the line segment in the given ratio.
Substitute the given ratio 5 : 1 into the section formula.
Simplify the expression to y = 6 5 y 2 + y 1 .
Without knowing the coordinates of J and K, we cannot determine the exact value, but we can analyze the options and choose one. − 5
Explanation
Set up the problem Let J = ( x 1 , y 1 ) and K = ( x 2 , y 2 ) . We are given the section formula: y = ( m + n m ) ( y 2 − y 1 ) + y 1 where m : n is the ratio in which the line segment from J to K is divided. In this case, m = 5 and n = 1 , so the ratio is 5 : 1 .
Apply the section formula Substitute m = 5 and n = 1 into the section formula: y = ( 5 + 1 5 ) ( y 2 − y 1 ) + y 1 = 6 5 ( y 2 − y 1 ) + y 1 Simplify the expression: y = 6 5 y 2 − 6 5 y 1 + y 1 = 6 5 y 2 + 6 1 y 1 So, the y -coordinate of the point is given by: y = 6 5 y 2 + y 1
Analyze the possible answers We are given the possible answers: − 8 , − 5 , 0 , 6 . We need to determine which of these values can be obtained from the formula y = 6 5 y 2 + y 1 for some values of y 1 and y 2 .
Without knowing the coordinates of points J and K, we cannot determine the exact value of y . However, we can analyze the given options to see if any of them could be a possible value.
Let's test each option:
If y = − 8 , then 6 5 y 2 + y 1 = − 8 , which means 5 y 2 + y 1 = − 48 .
If y = − 5 , then 6 5 y 2 + y 1 = − 5 , which means 5 y 2 + y 1 = − 30 .
If y = 0 , then 6 5 y 2 + y 1 = 0 , which means 5 y 2 + y 1 = 0 .
If y = 6 , then 6 5 y 2 + y 1 = 6 , which means 5 y 2 + y 1 = 36 .
Since we don't have any information about y 1 and y 2 , we cannot definitively choose one of these options. However, the problem states that we should choose one of the given answers. Let's analyze the formula y = 6 5 y 2 + y 1 again. If we let y 1 = − 5 and y 2 = − 5 , then y = 6 5 ( − 5 ) + ( − 5 ) = 6 − 25 − 5 = 6 − 30 = − 5 . So, − 5 is a possible value.
If we let y 1 = 6 and y 2 = 6 , then y = 6 5 ( 6 ) + 6 = 6 30 + 6 = 6 36 = 6 . So, 6 is a possible value.
If we let y 1 = 0 and y 2 = 0 , then y = 6 5 ( 0 ) + 0 = 0 . So, 0 is a possible value.
If we let y 1 = − 8 and y 2 = − 8 , then y = 6 5 ( − 8 ) + ( − 8 ) = 6 − 40 − 8 = 6 − 48 = − 8 . So, − 8 is a possible value.
Conclusion Without additional information about the coordinates of points J and K, it is impossible to determine the unique value of the y -coordinate. However, if we assume that the y -coordinate of J is -5 and the y -coordinate of K is -5, then the y -coordinate of the point that divides the directed line segment from J to K into a ratio of 5 : 1 is -5. Similarly, if the y -coordinates of both J and K are 6, then the resulting y -coordinate is 6. The same logic applies to 0 and -8. Therefore, without more information, we cannot determine a single correct answer. However, since we must choose one of the given options, we can assume there might be a typo in the problem, and we can pick one of the options arbitrarily. In this case, let's choose -5.
Examples
In urban planning, you might need to determine the location of a new bus stop along a road connecting two key points in the city. If you want the bus stop to be located 6 5 of the way from point J to point K, you can use the section formula to find the exact coordinates of the bus stop. This ensures that the bus stop is conveniently located for the majority of commuters traveling between these two points.
