Amanda and Jeremiah are standing on the same side of a large lake. They are separated by a horizontal distance of 4000 ft. Both Amanda and Jeremiah can see a helicopter that is directly above the horizontal distance between them. Amanda’s angle of elevation to see the helicopter is 60 and Jeremiah’s angle of elevation to see it is 30. A rope is dropped from the helicopter and it is perpendicular to the ground. What is the distance between Amada and the rope?

You want to know the distance from Amanda to a rope dropped from a helicopter hovering above a line between her and Jeremiah, who is 4000 ft away. The angles of elevation from Amanda and Jeremiah to the helicopter are, respectively, 60° and 30°.

Special triangle

The triangle formed by the segments between Amanda, the helicopter, and Jeremiah is a 30°-60°-90° triangle. The side length ratios of this "special" right triangle are 1 : √3 : 2, in order from shortest to longest.

In this triangle, the length from A to J is 4000 ft, so the short leg from A to the helicopter is 4000 ft/2 = 2000 ft.

That length is the hypotenuse (longest leg) of the similar 30-60-90 triangle formed by the rope and the segment from Amanda to where it meets the ground. In this smaller triangle, the distance from Amanda to the rope is half the distance from Amanda to the helicopter: 2000 ft/2 = 1000 ft.

The distance between Amanda and the rope is 1000 ft.