Q16. - Geometry

Question

Surveyors and photographers use a tripod for support.

Think of the intersection of the ceiling and the front wall of your classroom as line $\mathbf{I}$. Let the point in the center of the floor be point $\mathbf{C}$.

1. Is there a plane that contains line $\mathbf{I}$ and point $\mathbf{C}$?
2. State the theorem that applies.

Step-by-Step Solution

Answer

1. Yes, there is a plane that contain line $I$ and point $C$.
2. The theorem that applies is Theorem 1-2 which states that there is exactly one plane passing through a line and a point which is not on the line.

Part a. Step 1. Draw a cuboid representing the classroom.

The figure representing the classroom is:

Part a. Step 2. Observe the diagram.

The plane $PQRS$ is representing the front wall of the classroom.

The plane $PSFE$ is representing the ceiling of the classroom.

The planes $PQRS$ and $PSFE$ are intersecting on the line $PS$. Therefore, the line $PS$ is represented by the line $I$.

Part a. Step 3. Draw point C in the center of the floor and line l.

The figure representing the point $C$ in the center of the floor and line $I$ is:

Part a. Step 4. Construct a plane PSAB.

The figure representing the plane $PSAB$ is:

Part a. Step 5. State whether there is a plane that contains the line l and point C.

From the above diagram, it can be noticed that the plane $PSAB$ is passing through the line $I$ and point $C$.

Therefore, yes, there is a plane that is passing through the line $I$ and point $C$.

Part b. Step 1. Conditions to be satisfied.

The conditions to be satisfied to find which theorem applies is that there is one plane and that plane is passing through a line and a point that is not on the line.

Part b. Step 2. Write the Theorem 1-2.

There is exactly one plane passing through a line and a point that is not on the line.

Part b. Step 3. Name the theorem that applies to the given conditions.

The theorem that applies is Theorem 1-2 which states that there is exactly one plane passing through a line and a point that is not on the line.