What is the $y$-coordinate of the point that divides the directed line segment from J to K into a ratio of $5: 1$?
Answer (2)
Define the coordinates of points J and K as y 1 = − 8 and y 2 = 6 , respectively.
Apply the formula for dividing a line segment in a given ratio: y = ( m + n m ) ( y 2 − y 1 ) + y 1 , where m = 5 and n = 1 .
Substitute the given values into the formula: y = ( 5 + 1 5 ) ( 6 − ( − 8 )) + ( − 8 ) .
Simplify the expression to find the y -coordinate: y = 3 11 .
3 11
Explanation
Problem Analysis We are given two points, J and K, and we want to find the y -coordinate of the point that divides the directed line segment from J to K in the ratio 5 : 1 . We are given the formula v = ( m + n m ) ( v 2 − v 1 ) + v 1 , where v is the position vector of the point dividing the line segment in the ratio m : n . In this case, we are looking for the y -coordinate, so we will use the y -coordinates of points J and K.
Define Coordinates Let J = ( x 1 , y 1 ) and K = ( x 2 , y 2 ) . We are given that y 1 = − 8 and y 2 = 6 . We want to find the y -coordinate of the point that divides the segment JK in the ratio 5 : 1 .
Apply the Formula Using the given formula, the y -coordinate of the point is y = ( m + n m ) ( y 2 − y 1 ) + y 1 , where m = 5 and n = 1 .
Substitute Values Substitute the given values into the formula: y = ( 5 + 1 5 ) ( 6 − ( − 8 )) + ( − 8 ) .
Simplify Simplify the expression:
y = ( 6 5 ) ( 6 + 8 ) − 8
y = ( 6 5 ) ( 14 ) − 8
y = 6 70 − 8
y = 3 35 − 3 24
y = 3 11
y = 3 3 2 ≈ 3.67
Final Answer The y -coordinate of the point that divides the directed line segment from J to K in the ratio of 5 : 1 is 3 11 .
Examples
In urban planning, determining the location of a new bus stop along a street segment can use the concept of dividing a line segment in a given ratio. If city planners want to place the bus stop 6 5 of the way between two intersections, they can use this formula to find the exact coordinates for the bus stop, ensuring equitable access for residents.
The y -coordinate of the point that divides the directed line segment from J to K in a ratio of 5 : 1 is 3 11 . This is calculated using the formula that divides a segment based on a given ratio and the y -coordinates of J and K. Therefore, the answer is 3 11 .
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