In a regular triangular prism ABCA1B1C1, the side of the base is 4√3. E and F are the midpoints of edges A1B1

| March 17, 2021 | education

In a regular triangular prism ABCA1B1C1, the side of the base is 4√3. E and F are the midpoints of edges A1B1 and AC, respectively. Find the distance between lines AA1 and EF.

Through the straight line EF we draw a section EE1FF1 parallel to the face CC1B1B, and therefore parallel to the edge AA1. The perpendicular AH to the FF1 side of the section will be our value.

Consider a triangle AFF1, in which AF = AF1 = AC / 2, since points E and F are the midpoints of A1B1 and AC. The segment FF1 is the middle line of the triangle ABC, then FF1 = BC / 2, which means that the triangle AFF1 is equilateral, with a side of 4 * √3 / 2 = 2 * √3 cm.

AH is the height of an equilateral triangle. AH = FF1 * √3 / 2 = 2 * √3 * √3 / 2 = 3 cm.

Answer: The distance between the lines is 3 cm.



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