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# how to find the radius of a sphere?

Question:

The figure shows a right-angle prism. AC is perpendicular to the plane CBB1C1. AC = 4, BC = square root of 23. CC1 = 5. If all vertices of the right-angle prism lie on a sphere, then what is the radius of the sphere? Solution:

Draw a line passes the point A and parallel to CB, draw this line to point D, to make AD = CB. Connect DB.

Draw a line passes the point A1 and parallel to C1B1, draw this line to point D1, to make A1D1 = C1B1. Connect D1B1.

Connect to DD1

Now, ADD1A1-CBB1C1 is a right rectangle prism. Connect AB1, AB1 is the longest diagonal of the right rectangle prism.

Find the midpoint of AB1, name this point as point O. The distance from the point O to each vertex are equal. So, AB1 = 2R, R is the radius of the sphere. Now, we will find R.

In rectangle CBB1C1, the angle B = 90o. connect CB1.

CB12 = CB2 + BB12 = (square root of 23)2 + 52

Because AC is perpendicular to the plane CBB1C1, and CB1 lies on the plane CBB1C1. So, AC is perpendicular to CB1. So, triangle ACB1 is a right triangle.

In right triangle ACB1
AB12 = AC2 + CB12
= 42 + (square root of 23)2 + 52
= 64
AB1 = 8
2R = 8
R = 4

Therefore, when all vertices of the right-angle prism lie on a sphere, the radius of the sphere is 4. Watch the video for more details.