Dividing the King’s Foot into 12 Equal Pieces
Use a compass to mark off equally spaced points 𝐶, 𝐷, 𝐸, and 𝐹 so that 𝐴𝐵, 𝐵𝐶, 𝐶𝐷, 𝐷𝐸, and 𝐸𝐹 are equal in length. Describe the steps you took.
Exploratory Challenge 1
Divide segment 𝐴𝐵 into three segments of equal lengths
Divide segment 𝐴𝐵 into five segments of equal lengths
Exploratory Challenge 2
Divide segment 𝐴𝐵 into four segments of equal length.
On a piece of poster paper, draw a segment 𝐴𝐵 with a measurement of 1 foot. Use the dilation method to divide ̅𝐴𝐵̅̅̅ into twelve equal-length segments, or into 12 inches.
SIDE SPLITTER METHOD: If 𝐴𝐵 is a line segment, construct a ray 𝐴𝐴1, and mark off 𝑛 equally spaced points using a compass of fixed radius to get points 𝐴 = 𝐴0, 𝐴1, 𝐴2,⋯, 𝐴𝑛. Construct 𝐴𝑛𝐵 that is a side of △ 𝐴𝐵𝐴𝑛. Through each point 𝐴1, 𝐴2, …, 𝐴𝑛−1, construct 𝐴𝑖𝐵𝑖 parallel to 𝐴𝑛𝐵 that connect two sides of △ 𝐴𝐴𝑛𝐵.
DILATION METHOD: Construct a ray 𝑋𝑌 parallel to 𝐴𝐵. On the parallel ray, use a compass to mark off 𝑛 equally spaced points 𝑋1, 𝑋2,⋯, 𝑋𝑛 so that 𝑋𝑋𝑛 ≠ 𝐴𝐵. 𝐴𝑋 and 𝐵𝑋𝑛 intersect at a point 𝑂. Construct the rays 𝑂𝑋𝑖 that meet 𝐴𝐵 in points 𝐴𝑖.
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