1. **Problem statement:** We have three identical rectangles arranged in a stair-like formation on a coordinate plane. The bottom-left corner of the lowest rectangle is at $ (1,4) $ and the top-right corner of the highest rectangle is at $ (7,12) $. We need to find the coordinates of point $ H $, which lies on the vertical segment connecting the middle and top rectangles on the right side. 2. **Understanding the problem:** Since the rectangles are identical and arranged stair-like, each rectangle has the same width and height. The total horizontal distance from $ x=1 $ to $ x=7 $ covers the widths of all three rectangles, and the total vertical distance from $ y=4 $ to $ y=12 $ covers the heights of all three rectangles. 3. **Calculate the width and height of each rectangle:** - Total width $ = 7 - 1 = 6 $ - Total height $ = 12 - 4 = 8 $ - Since there are 3 identical rectangles, each rectangle's width $ w = \frac{6}{3} = 2 $ - Each rectangle's height $ h = \frac{8}{3} = \frac{8}{3} $ 4. **Find the coordinates of point $ H $:** - Point $ H $ lies on the right side vertical segment between the middle and top rectangles. - The right side of the rectangles is at $ x = 1 + 3 \times w = 1 + 3 \times 2 = 7 $ - The vertical segment between the middle and top rectangles is between $ y = 4 + 2h $ and $ y = 4 + 3h $ - Calculate these $ y $ values: $$ y_{bottom} = 4 + 2 \times \frac{8}{3} = 4 + \frac{16}{3} = \frac{12}{3} + \frac{16}{3} = \frac{28}{3} \approx 9.33 $$ $$ y_{top} = 4 + 3 \times \frac{8}{3} = 4 + 8 = 12 $$ - Point $ H $ is on this vertical segment, so its $ x $ coordinate is $ 7 $ and its $ y $ coordinate is $ y_{bottom} = \frac{28}{3} $ (the bottom of the vertical segment between middle and top rectangles). 5. **Final answer:** $$ H = \left(7, \frac{28}{3}\right) \approx (7, 9.33) $$