A satellite S is in a geosynchronous orbit; that is, it stays over the same point T on Earth as Earth rotates on its axis. From the satellite, a spherical cap of Earth is visible. The circle bounding this cap is called the horizon circle. The line of sight from the satellite is tangent to a point Q on the surface of Earth. If the radius of Earth, CQ, is 3963 miles and the satellite is 23300 miles form the surface of the Earth at Q, what is the radius of the horizon circle, PQ, in miles?
I've been staring at this problem for hours. I'm completely stumped.
Start by drawing a picture: draw a circle, representing the earth, and a point, representing the satellite. Draw lines from the point tangent to the circle, at Q and Q', representing the lines of sight. Finally, draw a line from the satellite to the center of the earth, and lines from the center of the earth to the points at which the tangent lines touch the circle. You now have two right triangles and you know the length of one leg, the radius of the earth, and the hypotenuse, the radius of the earth plus the height of the satellite, so you can calculate the distance from the satellite to Q and Q' and the angle, , in either of those triangles, at the satellite ( = "near side/hypotenuse".
Draw the line QQ', crossing the line from the center of the earth to the satellite at point P. That gives another pair of right triangles having angle and hypotenuse equal to the distance from the satellite to Q or Q'.
(earboth, whose post I did not see before writing this, has a very nice picture with his solution.)