Final answer:
To make the line segment NG perpendicular to HS, the slope of NG is -4, and the slope of HS must be 1/4. After calculating, the value of y for point H is determined to be 8.
Step-by-step explanation:
To find the value of y such that the line segment NG is perpendicular to HS, we need to use the concept of slope.
The slope between two points (x1, y1) and (x2, y2) is given by (y2 - y1) / (x2 - x1). Therefore, the slope for NG is (10 - (-2)) / (4 - 7) = 12 / -3 = -4.
For two lines to be perpendicular, the product of their slopes must be -1. If we let m represent the slope of HS, with H having coordinates (2, y) and S being (70, 5), then m * (-4) = -1.
The slope for HS is (5 - y) / (70 - 2) = (5 - y) / 68. Solving for y using the perpendicular condition, we have (-4) * ((5 - y) / 68) = -1, which simplifies to y = 8.