The radius of the ball is 12. The point is on the tangent plane and at a distance of 16

| June 2, 2021 | education

The radius of the ball is 12. The point is on the tangent plane and at a distance of 16 from the point of tangency. Find its shortest distance from the surface of the ball.

Let’s draw an axial section of the ball perpendicular to the tangent plane. Let’s complete the drawing.

Let K be the point of contact between the ball and the plane, point A belongs to the plane, AK = 16. The radius of the ball with center O is R = 12.

Let C be the point of intersection of the segment OA and the surface of the ball. AC will be the desired distance.

Consider a triangle OKA: angle K – 90 ° (since the radius is perpendicular to the tangent), OK = R = 12. AK = 16 (by condition). We calculate the length of OA by the Pythagorean theorem:

OA = √ (OK² + AK²) = √ (12² + 16²) = √ (144 + 256) = √400 = 20.

The OA segment consists of two segments OC and AC. ОC = R = 12. Find the length of АC:

AC = OA – OC = 20 – 12 = 8.

Answer: the distance from a point to the surface of the ball is 8.



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