A satellite is 6,000 miles from the horizon of Earth. Earth’s radius is about 4,000 miles. Find the approximate distance the satellite is from the point directly below it on Earth’s surface. The diagram is not to scale.

A right triangle can be used to model the geometry of the problem. One leg of it is the radius of the Earth. The leg at right angles to that is the satellite-to-horizon distance of 6000 miles. The hypotenuse of the triangle is the distance from the satellite to the center of the Earth, so the question will be answered by subtracting the Earth radius from that.

The Pythagorean theorem relates the various distances. Refer to the attachment.

AB² = AD² +BD²

(BC +4000)² = 4000² +6000² . . . . . . . . . . . . use given values

BC +4000 = 1000√(4² +6²) = 2000√13 . . . . take the square root

BC = 2000(√13 -2) . . . . . subtract 4000

BC ≈ 3211.1

The satellite is about 3211 miles from the point directly below it.