Q18. - Geometry

Question

Points $A$, $B$,  $C$ and $D$ are four non-coplanar points.

a. State the postulate that guarantees the existence of planes $ABC$, $ABD$, $ACD$, and $BCD$.

b. Explain how the ruler postulate guarantees the existence of a point $P$ between $A$ and $D$.

c. State the postulate the guarantees the existence of plane $BCP$.

d. Explain why there are an infinite number of planes through $\overline{BC}$.

Step-by-Step Solution

Answer

a. The postulate that guarantees the existence of planes $ABC$, $ABD$, $ACD$, and $BCD$ is postulate 7 which states that that through any three points there is at least one point and through any three non collinear points there is exactly one plane.

b. The ruler postulate states that the distance between any two points equals the absolute value of the difference of coordinates.

Consider that $P$ the point is between the points $A$ and $D$ having the coordinates as $x$, $y$ and $z$ respectively and also consider that $x<y<z$.

By using the ruler postulate it can be obtained that:

$\begin{array}{l}AP=\left|y-x\right|\\ PD=\left|z-y\right|\\ AD=\left|z-x\right|\end{array}$

As, $x<y<z$, therefore $y-x$ will be positive and similarly $z-y$ and $z-x$ will also be positive.

Therefore, it can be noticed that:

$\begin{array}{l}AP=y-x\\ PD=z-y\\ AD=z-x\end{array}$

Therefore, it can be obtained that:

$\begin{array}{c}AP+PD=y-x+z-y\\ =z-x\\ =AD\end{array}$

Therefore, $AP+PD=AD$.

Therefore, the point $P$ lies between the points $A$ and $D$.

Therefore, by using the ruler postulate it is proved that the point $P$ lies between the points $A$ and $D$.

c. The postulate that guarantees the existence of plane $BCP$ is postulate 7 which states that that through any three points there is at least one point and through any three non collinear points there is exactly one plane.

d. As, in the line $\overline{BC}$, there are infinite number of collinear points and through collinear points infinite number of planes can pass. Therefore, infinite number of planes can pass through $\overline{BC}$.

Part a. Step 1. Observe the given diagram.

The given diagram is:

Part a. 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 a. Step 3. State the postulate that guarantees the existence of planes A B C , A B D , A C D , and B C D .

As the points $A$, $B$, $C$ and $D$ are non collinear points therefore by using the postulate 7 it can be said that there is exactly one plane that can pass through these points.

Therefore, the postulate that guarantees the existence of planes $ABC$, $ABD$, $ACD$, and $BCD$ is postulate 7 which states that 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 1. Observe the given diagram.

The given diagram is:

Part b. Step 2. Write the Ruler postulate.

The Ruler postulate states that:

1. The points on a line can be paired with the real numbers in such a way that any two points can have coordinate 0 and 1.

2. Once a coordinate system has been chosen in this way, the distance between any two points equals the absolute value of the difference of their coordinates.

Part b. Step 3. Description of step .

The ruler postulate states that the distance between any two points equals the absolute value of the difference of coordinates.

Consider that the point $P$ is between the points $A$ and $D$ having the coordinates as $x$, $y$ and $z$ respectively and also consider that $x<y<z$.

By using the ruler postulate it can be obtained that:

$\begin{array}{l}AP=\left|y-x\right|\\ PD=\left|z-y\right|\\ AD=\left|z-x\right|\end{array}$

As, $x<y<z$, therefore $y-x$ will be positive and similarly $z-y$ and $z-x$ will also be positive.

Therefore, it can be noticed that:

$\begin{array}{l}AP=y-x\\ PD=z-y\\ AD=z-x\end{array}$

Therefore, it can be obtained that:

$\begin{array}{c}AP+PD=y-x+z-y\\ =z-x\\ =AD\end{array}$

Therefore, $AP+PD=AD$.

Therefore, the point $P$ lies between the points $A$ and $D$.

Therefore, by using the ruler postulate it is proved that the point $P$ lies between the points $A$ and $D$.

Part c. Step 1. Observe the given diagram.

The given diagram is:

Part c. 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 c. Step 3 - State the postulate that guarantees the existence of plane B C P .

As the points $B$, $C$ and $P$ are non collinear points therefore by using the postulate 7 it can be said that there is exactly one plane that can pass through these points.

Therefore, the postulate that guarantees the existence of plane $BCP$ is postulate 7 which states that 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 1.- Observe the given diagram.

The given diagram is:

Part d. Step 2. Description of step.

Through any two points, infinite number of planes can pass. To have a unique plane, three non collinear points are required.

Part d. Step 3. Description of step.

As, in the line $\overline{BC}$, there are infinite number of collinear points and through collinear points infinite number of planes can pass.

Therefore, infinite number of planes can pass through $\overline{BC}$.