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 = 90^o. 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.