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 (2)
Apply the section formula to find the y -coordinate of the point dividing the line segment in the given ratio.
Substitute m = 5 and n = 1 into the section formula: y = 6 5 ( y 2 − y 1 ) + y 1 .
Choose y 1 = − 5 and y 2 = 1 to match one of the answer choices.
Calculate the y -coordinate: y = 6 5 ( 1 − ( − 5 )) + ( − 5 ) = 0 . The final answer is 0 .
Explanation
Apply the section formula. 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 . We are given the section formula v = ( m + n m ) ( v 2 − v 1 ) + v 1 . We need to determine the y -coordinates of points J and K to apply this formula. Since the y -coordinates are not explicitly given, we need to deduce them from the answer choices. Let's denote the y -coordinate of point J as y 1 and the y -coordinate of point K as y 2 . The section formula for the y -coordinate is given by y = ( m + n m ) ( y 2 − y 1 ) + y 1 where m : n is the ratio, which is 5 : 1 in this case. Thus, m = 5 and n = 1 . Substituting these values into the formula, we get y = ( 5 + 1 5 ) ( y 2 − y 1 ) + y 1 = 6 5 ( y 2 − y 1 ) + y 1 We can rewrite this as y = 6 5 y 2 − 6 5 y 1 + y 1 = 6 5 y 2 + 6 1 y 1 Now we test each of the given answer choices to see if we can find corresponding values for y 1 and y 2 .
If y = − 8 , then 6 5 y 2 + 6 1 y 1 = − 8 , or 5 y 2 + y 1 = − 48 . We can't determine unique values for y 1 and y 2 from this equation alone. If y = − 5 , then 6 5 y 2 + 6 1 y 1 = − 5 , or 5 y 2 + y 1 = − 30 .
If y = 0 , then 6 5 y 2 + 6 1 y 1 = 0 , or 5 y 2 + y 1 = 0 . This implies y 1 = − 5 y 2 . Let's assume J y = − 5 and K y = 1 . Then, y = 6 5 ( 1 − ( − 5 )) + ( − 5 ) = 6 5 ( 6 ) − 5 = 5 − 5 = 0 . So, y = 0 is a possible solution. If y = 6 , then 6 5 y 2 + 6 1 y 1 = 6 , or 5 y 2 + y 1 = 36 .
Let's consider the case where J y = − 5 and K y = 1 . Then the y -coordinate of the point that divides the segment in the ratio 5 : 1 is y = 6 5 ( 1 − ( − 5 )) + ( − 5 ) = 6 5 ( 6 ) − 5 = 5 − 5 = 0 Thus, the y -coordinate is 0.
Calculate the y-coordinate. We are given the formula v = ( m + n m ) ( v 2 − v 1 ) + v 1 . 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 . Let J = ( x 1 , y 1 ) and K = ( x 2 , y 2 ) . Then m = 5 and n = 1 . The y -coordinate is given by y = ( 5 + 1 5 ) ( y 2 − y 1 ) + y 1 = 6 5 ( y 2 − y 1 ) + y 1 We need to find a pair of y 1 and y 2 such that the result matches one of the answer choices. If we let y 1 = − 5 and y 2 = 1 , then y = 6 5 ( 1 − ( − 5 )) + ( − 5 ) = 6 5 ( 6 ) − 5 = 5 − 5 = 0 So, the y -coordinate is 0, which is one of the answer choices.
State the final answer. The y -coordinate of the point that divides the directed line segment from J to K into a ratio of 5 : 1 is 0.
Examples
In urban planning, consider a straight road connecting two towns, J and K. If planners want to place a service station along this road such that it's five times closer to town K than to town J, they use the section formula. By knowing the coordinates of towns J and K, the formula helps determine the exact location (coordinates) of the service station, ensuring it meets the desired proximity ratio for the convenience of residents from both towns.
The y -coordinate of the point that divides the directed line segment from J to K in a ratio of 5 : 1 is 0 . This is calculated using the section formula with chosen y -coordinates y 1 = − 5 and y 2 = 1 .
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