Q4. - Geometry

Question

Plane Z passes through the center of sphere Q.

1. Explain why $QR=QS=QT$.
2. Explain why the intersection of the plane and the sphere is a circle. (The intersection of a sphere with any plane passing through the center of the sphere is called a great circle of the sphere).

Step-by-Step Solution

Answer

a. $QR=QS=QT$. 

b. The intersection of a plane and a sphere is a circle. 

a . Step 1. Given:

Sphere is cut by a plane Z.

Step 2. Explanation:

In sphere all points on surface are equidistance from the centre when the plane cuts the sphere. Since all three given points R, S, T lie on the surface of the sphere, they are also equidistance from the point, which is the centre of the sphere.

Therefore,  $QR=QS=QT$

b. Step 1. Given:

Sphere is cut by a plane Z.

Step 2. Explanation:

Since we know that intersection of any plane passing through the centre of the sphere form a great circle and sphere is a three dimensional symmetric version of a circle (with varying radius), he the figure so formed by the intersection of a plane and a sphere is always a circle.

Therefore, The intersection of a plane and a sphere is a circle.