If the diameter of the Earth is approximately 12,800 km, how far is the astronaut from the Space Center?

Question

<!--[if !supportLists]-->1.    <!--[endif]-->The average height of the International Space Station is 400km above the Earth. An astronaut on board the International Space Station sights along a tangent line to Earth, viewing Houston Space Center. If the diameter of the Earth is approximately 12,800 km, how far is the astronaut from the Space Center. Draw/Illustrate the problem

Answer 1

A right triangle is formed by joining the space station to the centre of the earth (400+6400=6800km). The tangent point is distance 6400km from the centre of the earth (radius) and the line of sight makes a right angle with the radius. So the distance between the space station and the tangent point is √(6800²-6400²)=√5280000=2297.8km approx.

In the picture the blue circle represents the earth, point A is the astronaut at the space station and H represents the location of Houston. AH is the distance required, OH is a radius=12800/2=6400km. AO=6400+400=6800km (not to scale). ∠AHO is a right angle.