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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?

find radius of the sphere if all vertices of a right angle prism lie on a 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.

find radius of a sphere when all vertices of a right angle prism lie on a sphere.

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.