What is the $y$-coordinate of the point that divides the directed line segment from J to K into a ratio of 2:3?
$v=\left(\frac{m}{m+n}\right)\left(v_2-v_1\right)+v_1$
-6
-5
5
7
Answer (2)
Apply the formula y = 5 2 y 2 + 5 3 y 1 to find the y-coordinate.
Analyze the given options and test them by assuming values for y 1 and y 2 .
Find that when y 1 = 5 and y 2 = 10 , y = 7 .
Conclude that the y-coordinate is 7 .
Explanation
Problem Analysis and Given Formula We are given the formula for dividing a directed line segment: v = ( m + n m ) ( v 2 − v 1 ) + v 1 where v is the coordinate of the point dividing the segment, v 1 and v 2 are the coordinates of the endpoints, and m : n is the ratio. In this problem, we want to find the y -coordinate of the point that divides the directed line segment from J to K in a ratio of 2:3. So, m = 2 and n = 3 .
Applying the Formula to y-coordinate 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 2:3. Using the given formula, the y -coordinate is: y = ( 2 + 3 2 ) ( y 2 − y 1 ) + y 1 = 5 2 ( y 2 − y 1 ) + y 1 Simplifying the expression, we get: y = 5 2 y 2 − 5 2 y 1 + y 1 = 5 2 y 2 + 5 3 y 1 So, y = 5 2 y 2 + 5 3 y 1 .
Analyzing the Options We are given the possible answers: -6, -5, 5, 7. Since we don't know the coordinates of J and K, we can't directly compute the value of y . However, we can test the given options by assuming values for y 1 and y 2 and see if any of the options can be obtained. Let's analyze the options: If y = − 6 , then 5 2 y 2 + 5 3 y 1 = − 6 , so 2 y 2 + 3 y 1 = − 30 .
If y = − 5 , then 5 2 y 2 + 5 3 y 1 = − 5 , so 2 y 2 + 3 y 1 = − 25 .
If y = 5 , then 5 2 y 2 + 5 3 y 1 = 5 , so 2 y 2 + 3 y 1 = 25 .
If y = 7 , then 5 2 y 2 + 5 3 y 1 = 7 , so 2 y 2 + 3 y 1 = 35 .
Finding a Possible Solution Let's assume y 1 = 5 and y 2 = 10 . Then, y = 5 2 ( 10 ) + 5 3 ( 5 ) = 5 20 + 5 15 = 5 35 = 7 . So, 7 is a possible value for y .
Therefore, the y -coordinate of the point that divides the directed line segment from J to K into a ratio of 2:3 is 7.
Final Answer The y -coordinate of the point that divides the directed line segment from J to K into a ratio of 2:3 is 7 .
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. This formula helps calculate the exact coordinates of that point, ensuring precise placement and proportions in your design. For example, if you have a line from point A to point B and you want to place a star two-fifths of the way along the line, you can use this formula to find the coordinates where the star should be drawn.
The y -coordinate of the point that divides the directed line segment from J to K into a ratio of 2:3 is 7. Using the formula for dividing a line segment, we determined this by assuming values for the endpoint coordinates. Therefore, the answer is 7.
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