# SAT II Math II : 3-Dimensional Axes and Coordinates

## Example Questions

### Example Question #1 : 3 Dimensional Axes And Coordinates

Which of the following numbers comes closest to the length of line segment in three-dimensional coordinate space whose endpoints are the origin and the point  ?

Explanation:

Use the three-dimensional version of the distance formula:

The closest of the five choices is 7.

### Example Question #111 : Geometry

A line segment  in three-dimensional space has midpoint  has midpoint .

has Cartesian coordinates  has Cartesian coordinates . Give the -coordinate of .

Explanation:

The midpoint formula for the -coordinate

will be applied twice, once to find the -coordinate of , then again to find that of .

First, set , the -coordinate of , and , the -coordinate of , and solve for , the -coordinate of :

Now, set , the  -coordinate of , and , the -coordinate of , and solve for , the -coordinate of :

### Example Question #3 : 3 Dimensional Axes And Coordinates

A line segment in three-dimensional space has endpoints with Cartesian coordinates  and . To the nearest tenth, give the length of the segment.

Explanation:

Use the three-dimensional version of the distance formula:

### Example Question #4 : 3 Dimensional Axes And Coordinates

A pyramid is positioned in three-dimensional space so that its four vertices are located at the points with coordinates , and the origin. Give the volume of this pyramid.

Explanation:

The three segments that connect the origin to the other points are all contained in one of the -, -, and - axes. Thus, this figure can be seen as a pyramid with, as its base, a right triangle in the -plane with vertices , and the origin, and, as its altitude, the segment with the origin and  as its endpoints.

The segment connecting the origin and  is one leg of the base and has length 6; the segment connecting the origin and  is the other leg of the base and has length 9; the area of the base is therefore

The segment connecting the origin and  is the altitude; its length - the height of the pyramid - is 12.

The volume of the pyramid is

### Example Question #5 : 3 Dimensional Axes And Coordinates

A pyramid is positioned in three-dimensional space so that its four vertices are located at the points with coordinates , and the origin. Give the volume of this pyramid.

Explanation:

The three segments that connect the origin to the other points are all contained in one of the -, -, and - axes. Thus, this figure can be seen as a pyramid with, as its base, a right triangle in the -plane with vertices , and the origin, and, as its altitude, the segment with the origin and  as its endpoints.

The segment connecting the origin and  is one leg of the base and has length ; the segment connecting the origin and is the other leg of the base and has length ; the area of the base is therefore

The segment connecting the origin and  is the altitude; its length - the height of the pyramid - is .

The volume of the pyramid is

### Example Question #1 : 3 Dimensional Axes And Coordinates

A line segment  in three-dimensional space has midpoint  has midpoint .

has Cartesian coordinates  has Cartesian coordinates . Give the -coordinate of .

Explanation:

The midpoint formula for the -coordinate

will be applied twice, once to find the  -coordinate of , then again to find that of .

First, set , the  -coordinate of , and , the -coordinate of , and solve for , the -coordinate of :

Now, set , the  -coordinate of , and , the -coordinate of , and solve for , the -coordinate of :

### Example Question #1 : 3 Dimensional Axes And Coordinates

A line segment  in three-dimensional space has midpoint  has midpoint .

has Cartesian coordinates  has Cartesian coordinates . Give the -coordinate of .

Explanation:

The midpoint formula for the -coordinate

will be applied twice, once to find the  -coordinate of , then again to find that of .

First, set , the  -coordinate of , and , the -coordinate of , and solve for , the -coordinate of :

Now, set , the  -coordinate of , and , the -coordinate of , and solve for , the -coordinate of :