Q20. - Geometry

Question

Parts (a) through (d) justify theorem 1-2: through a line and a point not in the line there is exactly one plane.

a. If $P$ is a point not in line $k$, what postulate permits us to state that there are two points $R$ and $S$ in line $k$?

b. Then there is at least one plane $X$ that contains points $P$, $R$ and $S$. Why?

c. What postulate guarantees that plane $X$ contains line $k$? Now we know that there is a plane $X$ that contains both point $P$ and line $k$.

d. There can’t be another plane that contains point $P$ and line $k$, because then two planes would contain non collinear points  $P$, $R$ and $S$. What postulate does this contradict?

Step-by-Step Solution

Answer

a. The postulate that permits to say that points $R$ and $S$ lie in line is postulate 5.

b. Then there is at least one plane $X$ that contains points $P$, $R$ and $S$ because as the points $P$, $R$ and $S$ are non collinear points and by using the postulate 7 it can be said that through any three non collinear points $P$, $R$ and $S$ there is exactly one plane.

c. The postulate that guarantees that the plane $X$ contains line $k$ is postulate 8.

d. The postulate that contradicts the given statement is postulate 7.

Part a. Step 1. Observe the given diagram.

The given diagram is:

Part a. Step 2. Write the postulate 5.

The postulate 5 states that a line contains at least two points, a plane contains at least three points not all in one line.

Part a. Step 3. Description of step.

By using the postulate 5 it can be said that the line contains at least two points.

Therefore, the line $k$ contains at least two points $R$ and $S$.

Therefore, the postulate that permits to say that points $R$ and $S$ lie in line $k$ is postulate 5.

Part b. Step 1. Observe the given diagram.

The given diagram is:

Part b. Step 2. Write the postulate 7.

The postulate 7 states that through any three points there is at least one point and through any three non collinear points there is exactly one plane.

Part b. Step 3. Description of step .

Then there is at least one plane $X$ that contains points $P$, $R$ and $S$ because as the points $P$, $R$ and $S$ are non collinear points and by using the postulate 7 it can be said that through any three non collinear points $P$, $R$ and $S$ there is exactly one plane.

Part c. Step 1. Observe the given diagram.

The given diagram is:

Part c. Step 2. Write the postulate 8.

The postulate 8 states that if two points are in a plane, then the line that contains the points is in that plane.

Part c. Step 3. Description of step .

As the points $R$ and $S$ lies on the plane $X$, therefore by using the postulate 8 it can be said the line joining the points $R$ and $S$ will also lie on the plane $X$.

Therefore, the postulate that guarantees that the plane $X$ contains line $k$ is postulate 8.

Part d. Step 1. Observe the given diagram.

The given diagram is:

Part d. Step 2. Write the postulate 7.

The postulate 7 states that through any three points there is at least one point and through any three non collinear points there is exactly one plane.

Part d. Step 3. Description of step.

The postulate that contradicts the given statement is postulate 7.