Two mutually perpendicular sections are drawn through a point inside the ball. Find the distance between

Two mutually perpendicular sections are drawn through a point inside the ball. Find the distance between their centers if the area of each section is 9pi and the radius of the ball is 5.

Knowing the cross-sectional areas, we determine their radii.

S = n * R ^ 2 = n * 9.

R ^ 2 = 9.

R = 3 cm.

From the point O, the center of the ball, we draw the radii OS and OD and the perpendiculars OA and OB to the section planes.

Triangles AOD and OBC are rectangular, in which AD = BC = 3 cm, OD = OS = 5 cm, as the radii of the sections and the sphere. Then the triangles are equal in leg and hypotenuse.

From the right-angled triangle AOD, according to the Pythagorean theorem, OA ^ 2 = OD ^ 2 – AD ^ 2 = 25 – 9 = 16.

ОА = ОВ = 4 cm.

The quadrilateral AOBE is a square, since the angles BОА = BEA = 90, and ОВ = ОА.

Then AB ^ 2 = 2 * OA ^ 2 = 2 * 16 = 32.

AB = √32 = 4 * √2 cm.

Answer: The distance between the centers of the sections is 4 * √2 cm.