## SAT SUBJECT TEST MATH LEVEL 2

## PART 2

## REVIEW OF MAJOR TOPICS

## CHAPTER 2

Geometry and Measurement

2.2 Three-Dimensional Geometry

### COORDINATES IN THREE DIMENSIONS

The coordinate plane can be extended by adding a third axis, the *z*-axis, which is perpendicular to the other two. Picture the corner of a room. The corner itself is the origin. The edges between the walls and the floor are the *x-* and *y*-axes. The edge between the two walls is the *z*-axis. The first octant of this three-dimensional coordinate system and the point (1,2,3) are illustrated below.

A point that has zero as any coordinate must lie on the plane formed by the other two axes. If two coordinates of a point are zero, then the point lies on the nonzero axis. The three-dimensional Pythagorean theorem yields a formula for the distance between two points (*x*_{1},*y*_{1},*z*_{1}) and (*x*_{2},*y*_{2},*z*_{2}) in space:

A three-dimensional coordinate system can be used to graph the variable *z* as a function of the two variables *x* and *y*, but such graphs are beyond the scope of the Level 2 Test.

**EXAMPLES**

**1. The distance between two points in space A (2, y, –3) and B (1, –1, 4) is 9. Find the possible values of y .**

Use the formula for the distance between two points and set this equal to 9:

Square both sides and simplify to get (*y* + 1)^{2} = 31. Therefore, *y* 4.6 or *y* –6.6.

A sphere is the set of points in space that are equidistant from a given point. If the given point is (*a*, *b*, *c*) and the given distance is *r*, an equation of the sphere is (*x* – *a*)^{2} + (*y* – *b*)^{2} + (*z* – *c*)^{2} = *r*^{2}.

**2. Describe the graph of the set of points ( x, y, z) where **

(x – 6)^{2} + (y + 3)^{2} + (z – 2)^{2} = 36.

This equation describes the set of points whose distance from (6, –3, 2) is 6. This is a sphere of radius 6 with center at (6, –3, 2).

If only two of the variables appear in an equation, the equation describes a planar figure. The third variable spans the entire number line. The resulting three-dimensional figure is a solid that extends indefinitely in both directions parallel to the axis of the variable that is not in the equation, with cross sections congruent to the planar figure.

**3. Describe the graph of the set of points ( x, y, z) where x^{2} + z^{2} = 1.**

Since *y* is not in the equation, it can take any value. When restricted to the *xz* plane, the equation is that of a circle with radius 1 and center at the origin. Therefore in *xyz* space, the equation represents a cylindrical shape, centered at the origin, that extends indefinitely in both directions along the *y*-axis.

A solid figure can also be obtained by rotating a plane figure about some line in the plane that does not intersect the figure.

**4. If the segment of the line y = –2x + 2 that lies in quadrant I is rotated about the y-axis, a cone is formed. What is the volume of the cone?**

As shown in the figure below on the left, the part of the segment that lies in the first quadrant and the axes form a triangle with vertices at (0,0), (1,0), and (0,2). Rotating this triangle about the *y*-axis generates the cone shown in the figure below on the right.

The radius of the base is 1, and the height is 2. Therefore, the volume is

**EXERCISES**

1. The distance between two points in space, *P* (*x*,–1,–1) and *Q* (3,–3,1), is 3. Find the possible values of *x* .

(A) 1 or 2

(B) 2 or 3

(C) –2 or –3

(D) 2 or 4

(E) –2 or –4

2. The point (–4,0,7) lies on the

(A) *y*-axis

(B) *xy* plane

(C) *yz* plane

(D) *xz* plane

(E) *z* -axis

3. The region in the first quadrant bounded by the line 3*x* + 2*y* = 7 and the coordinate axes is rotated about the *x*-axis. What is the volume of the resulting solid?

(A) 8 units^{3}

(B) 20 units^{3}

(C) 30 units^{3}

(D) 90 units^{3}

(E) 120 units^{3}