Q17. - Geometry

Question

Points $R$, $S$ and $T$ are non collinear points.

1. State the postulate that guarantees the existence of a plane $X$ that contains $R$, $S$ and $T$.
2. Draw a diagram showing plane $X$ containing the non collinear points $R$, $S$ and $T$.
3. Suppose that $P$ is any point of $\overleftrightarrow{RS}$ other than $R$ and $S$. Does point $P$ lie in plane $X$? Explain.
4. State the postulate that guarantees that $\overleftrightarrow{TP}$ exists.
5. State the postulate that guarantees that $\overleftrightarrow{TP}$ is in plane $X$.

 

Step-by-Step Solution

Answer

a. The postulate that guarantees the existence of a plane $X$ that contains $R$, $S$ and $T$ is postulate 7.

b. The diagram showing plane $X$ containing the non collinear points $R$, $S$ and $T$ is:

c. Yes, the point $P$ lies in the plane $X$ because the points $R$ and $S$ are lying on the plane $X$ and therefore by using the postulate 8, it can be noticed that the line joining the points $R$ and $S$ will also lie on the plane $X$ and as the point $P$ is a point lying on the line joining the points $R$ and $S$. Therefore, consequently the point $P$ is a point lying on the plane $X$.

d. The postulate that guarantees that $\overleftrightarrow{TP}$ exists is postulate 6.

e. The postulate that guarantees that $\overleftrightarrow{TP}$ is in plane $X$ is postulate 8.

Part a. Step 1. Write the postulate 7.

The postulate 7 states that through any three points at least one plane passes and if the three points are non collinear then exactly one plane pass through these non collinear points.

Part a. Step 2. Draw a diagram having plane X and points R , S and T on the plane X .

The figure representing the plane $X$ having points $R$, $S$ and $T$ on the plane $X$ is:

Part a. Step 3. State the postulate that guarantees the existence of plane X that contains R , S and T .

From the postulate 7 and the above diagram it can be concluded that:

If there are three non collinear points then exactly one plane can pass through these non collinear points.

Therefore, the postulate that guarantees the existence of plane $X$ that contains $R$, $S$ and $T$ is postulate 7.

Part b. Step 1. Definition of non collinear points.

The non collinear points are the points which do not lie on the same line.

Part b. Step 2. Write the postulate 7.

The postulate 7 states that through any three points at least one plane passes and if the three points are non collinear then exactly one plane pass through these non collinear points.

Part b. Step 3. Draw a diagram having plane X and points R , S and T on the plane X .

The figure representing the plane $X$ having points $R$, $S$ and $T$ on the plane $X$ is:

Part c. Step 1. Write the postulate 8.

The postulate 8 states that if there are two points lying on the plane then the line joining these two points will also lie on that plane.

Part c. Step 2. Relation of the point P with the points R and S .

The point $P$ is a point lying on the line joining the points $R$ and $S$.

Part c. Step 3. Relation of the point P with the plane X .

The points $R$ and $S$ are lying on the plane $X$ and therefore by using the postulate 8, it can be noticed that the line joining the points $R$ and $S$ will also lie on the plane $X$.

As the point $P$ is a point lying on the line joining the points $R$ and $S$. Therefore, consequently the point $P$ is a point lying on the plane $X$.

Therefore, yes the point $P$ lies on the plane $X$.

Part d. Step 1. Write the postulate 6.

The postulate 6 states that if there any two points, then exactly one line can pass through these two points.

Part d. Step 2. Draw a diagram of T P ↔ .

The figure representing $\overleftrightarrow{TP}$ is:

Part d. Step 3. State the postulate that guarantees that T P ↔ exists.

The postulate that guarantees that $\overleftrightarrow{TP}$ exists is postulate 6.

Part e. Step 1. Write the postulate 8.

The postulate 8 states that if there are two points lying on the plane then the line joining these two points will also lie on that plane.

Part e. Step 2. Relation of the point P and T with the plane X .

The points $P$ and $T$ are the points lying on the plane $X$. 

Part e. Step 3. Relation of the point P with the plane X .

The points $P$ and $T$ are lying on the plane $X$ and therefore by using the postulate 8, it can be noticed that the line joining the points $P$ and $T$ will also lie on the plane $X$.

Therefore, the line $\overleftrightarrow{TP}$ will lie on the plane $X$.

Therefore, the postulate that guarantees that $\overleftrightarrow{TP}$ is in plane $X$ is postulate 8.