SAT II Math II : 3-Dimensional Geometry

Study concepts, example questions & explanations for SAT II Math II

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Example Questions

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Example Question #1 : 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.

Possible Answers:

Correct answer:

Explanation:

Use the three-dimensional version of the distance formula:

Example Question #1 : 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.

Possible Answers:

Correct answer:

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 #1 : 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.

Possible Answers:

Correct answer:

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 #2 : 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 .

Possible Answers:

Correct answer:

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 #7 : 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 .

Possible Answers:

Correct answer:

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 #41 : 3 Dimensional Geometry

A convex polyhedron has twenty faces and thirty-six vertices. How many edges does it have?

Possible Answers:

Correct answer:

Explanation:

The number of vertices , edges , and faces  of any convex polyhedron are related by By Euler's Formula:

Setting  and solving for :

The polyhedron has 54 edges.

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