What is the $y$-coordinate of the point that divides the directed line segment from $J$ to $K$ into a ratio of 5:17?
Answer (2)
Analyze the problem and identify the section formula as the key to finding the y-coordinate.
Apply the section formula y = 22 5 ( y 2 − y 1 ) + y 1 to all possible pairs of given y-coordinates.
Calculate the resulting y-coordinate for each pair.
Since none of the calculated y-coordinates match the given options, conclude that no solution exists within the provided options. No solution found within the given options.
Explanation
Problem Analysis The problem asks us to find the y -coordinate of a point that divides a directed line segment from point J to point K in the ratio 5:17. We are given the formula v = ( m + n m ) ( v 2 − v 1 ) + v 1 , where m = 5 and n = 17 . The possible y -coordinates are -8, -5, 0, and 6. We need to test all possible pairs of these values as the y -coordinates of points J and K and see if the resulting y -coordinate of the dividing point is among the given options.
Applying the Section Formula Let's denote the y -coordinate of point J as y 1 and the y -coordinate of point K as y 2 . The formula for the y -coordinate of the dividing point is y = ( 5 + 17 5 ) ( y 2 − y 1 ) + y 1 = 22 5 ( y 2 − y 1 ) + y 1 . We will now test all possible pairs of y 1 and y 2 from the given values -8, -5, 0, and 6.
Testing All Pairs We will iterate through all possible pairs of y 1 and y 2 and calculate the resulting y -coordinate. If the calculated y -coordinate is equal to one of the given options, we have found our answer.
If y 1 = − 8 and y 2 = − 5 , then y = 22 5 ( − 5 − ( − 8 )) + ( − 8 ) = 22 5 ( 3 ) − 8 = 22 15 − 8 = 22 15 − 176 = 22 − 161 ≈ − 7.318 . This is not in the given options.
If y 1 = − 8 and y 2 = 0 , then y = 22 5 ( 0 − ( − 8 )) + ( − 8 ) = 22 5 ( 8 ) − 8 = 22 40 − 8 = 11 20 − 8 = 11 20 − 88 = 11 − 68 ≈ − 6.182 . This is not in the given options.
If y 1 = − 8 and y 2 = 6 , then y = 22 5 ( 6 − ( − 8 )) + ( − 8 ) = 22 5 ( 14 ) − 8 = 22 70 − 8 = 11 35 − 8 = 11 35 − 88 = 11 − 53 ≈ − 4.818 . This is not in the given options.
If y 1 = − 5 and y 2 = − 8 , then y = 22 5 ( − 8 − ( − 5 )) + ( − 5 ) = 22 5 ( − 3 ) − 5 = 22 − 15 − 5 = 22 − 15 − 110 = 22 − 125 ≈ − 5.682 . This is not in the given options.
If y 1 = − 5 and y 2 = 0 , then y = 22 5 ( 0 − ( − 5 )) + ( − 5 ) = 22 5 ( 5 ) − 5 = 22 25 − 5 = 22 25 − 110 = 22 − 85 ≈ − 3.864 . This is not in the given options.
If y 1 = − 5 and y 2 = 6 , then y = 22 5 ( 6 − ( − 5 )) + ( − 5 ) = 22 5 ( 11 ) − 5 = 22 55 − 5 = 2 5 − 5 = 2.5 − 5 = − 2.5 . This is not in the given options.
If y 1 = 0 and y 2 = − 8 , then y = 22 5 ( − 8 − 0 ) + 0 = 22 5 ( − 8 ) = 22 − 40 = 11 − 20 ≈ − 1.818 . This is not in the given options.
If y 1 = 0 and y 2 = − 5 , then y = 22 5 ( − 5 − 0 ) + 0 = 22 5 ( − 5 ) = 22 − 25 ≈ − 1.136 . This is not in the given options.
If y 1 = 0 and y 2 = 6 , then y = 22 5 ( 6 − 0 ) + 0 = 22 5 ( 6 ) = 22 30 = 11 15 ≈ 1.364 . This is not in the given options.
If y 1 = 6 and y 2 = − 8 , then y = 22 5 ( − 8 − 6 ) + 6 = 22 5 ( − 14 ) + 6 = 22 − 70 + 6 = 11 − 35 + 6 = 11 − 35 + 66 = 11 31 ≈ 2.818 . This is not in the given options.
If y 1 = 6 and y 2 = − 5 , then y = 22 5 ( − 5 − 6 ) + 6 = 22 5 ( − 11 ) + 6 = 22 − 55 + 6 = − 2.5 + 6 = 3.5 . This is not in the given options.
If y 1 = 6 and y 2 = 0 , then y = 22 5 ( 0 − 6 ) + 6 = 22 5 ( − 6 ) + 6 = 22 − 30 + 6 = 11 − 15 + 6 = 11 − 15 + 66 = 11 51 ≈ 4.636 . This is not in the given options.
Conclusion Since none of the calculated y -coordinates match the given options, there might be an error in the problem statement or the given options. However, based on the provided information and the section formula, no solution exists within the given set of options.
Examples
In computer graphics, when drawing a line between two points on a screen, you might want to find the coordinates of a point that divides the line into a specific ratio. This is useful for creating effects like gradients or for placing objects along a path. The section formula helps determine the exact coordinates of that point, ensuring precise placement and visual accuracy in the graphical representation.
To find the y -coordinate of the point dividing the line segment from J to K in a ratio of 5:17, use the section formula y = 22 5 ( y 2 − y 1 ) + y 1 . Substitute the actual y -coordinates of J and K into the formula to find the desired y -coordinate. Repeat calculations with all potential y 1 and y 2 to check against given options.
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