# The ƒᴀʀтhest distance a satellite signal can directly reach is the length of the segment tangent to the curve?

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The ƒᴀʀтhest distance a satellite signal can directly reach is the length of the segment tangent to the curve of Earth’s surface. If the angle formed by the tangent satellite signals is 126°, what is the measure of the intercepted arc on Earth? The figure is not drawn to scale

• Assume the Earth is spherical.

Draw a circle, representing a great-circle section of the Earth, with the satellite at a point, S, above it. Draw the tangents from S to the circle, and from tangent points, A and B, to the center, C.

Then AC and BC are radii, and

SA ⊥ AC and SB ⊥ BC

∠ASB = 126º, making the angular measure of the intercepted arc:

∠ACB = 180º – 126º = 54º = 3π/10

(because the 4 angles of the quadrilateral CASB, which is a kite, must sum to 360º, and the other two of them, SAC and SBC, are right angles)

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