### All HiSET: Math Resources

## Example Questions

### Example Question #1 : Rotations

What is the result of rotating the point about the origin in the plane by ?

**Possible Answers:**

**Correct answer:**

Rotating a point

geometrically in the plane about the origin is equivalent to negating the coordinates of the point algebraically to obtain

.

Thus, since our initial point was

we negate both coordinates to get

as the rotation about the origin by .

### Example Question #1 : Rotations

Examine the figures in the above diagram. Figure 2 is the result of performing which of the following transformations on Figure 1?

**Possible Answers:**

**Correct answer:**

The diagram below superimposes the two figures:

The transformation moves the black diagonal to the position of the red diagonal, and, consequently, points and to points and , respectively. This constitutes two-tenths of a complete turn clockwise, or a clockwise rotation of

### Example Question #1 : Rotations

Rotate the above figure counterclockwise. Which figure is the result?

**Possible Answers:**

None of the other choices gives the correct result.

**Correct answer:**

A counterclockwise rotation of is ofa complete rotation. Observe the following diagram:

In the right figure, the question mark has been turned one-eighth of a complete turn counterclockwise. This is the correct orientation.

### Example Question #1 : Rotations

Let and be the midpoints of and , respectively.

Rotate the above hexagon clockwise, then reflect it about the line through . Call the image of after these transformations.

will be located in the same position as which of the following points?

**Possible Answers:**

**Correct answer:**

A rotation is equivalent to of a complete rotation, so rotate as follows:

The image of under this rotation, which we will call , is at .

Now, locate the midpoints and , and construct the line as described and shown below. Perform the reflection:

It can be seen that the image of under this transformation - the desired - is located at .

### Example Question #5 : Rotations

In the above octagon, let and be the midpoints of and , respectively.

Rotate the above octagon counterclockwise, then reflect it about . Call the image of after these transformations.

will be located in the same position as which of the following points?

**Possible Answers:**

**Correct answer:**

A rotation is equivalent to of a complete rotation, so rotate as follows:

The image of under this rotation, which we will call , is at .

Now, locate the midpoints and , and construct the line as described and shown below. Perform the reflection:

The image of under this transformation - the desired - is located at .

### Example Question #21 : Understand Transformations In The Plane

Rotate the above hexagon counterclockwise, then reflect it about . Call the image of after these transformations.

will be located in the same position as which of the following points?

**Possible Answers:**

**Correct answer:**

A rotation is equivalent to of a complete rotation, so rotate as follows:

The image of under this rotation, which we will call , is at .

Now, construct , and reflect the hexagon about this line:

The image of under this reflection - the desired - is located at itself.

### Example Question #1 : Rotations

Between 6:15 and 6:40, the minute hand of a clock undergoes which of the following clockwise rotations?

**Possible Answers:**

**Correct answer:**

Between 6:15 and 6:40,

minutes elapse.

The minute hand of a clock rotates one complete clockwise turn about its mount over the course of 60 minutes. Therefore, over 25 minutes, the minute hand rotates clockwise.

### Example Question #21 : Understand Transformations In The Plane

Over the course of 20 minutes, the hour hand of a clock undergoes which of the following rotations?

**Possible Answers:**

**Correct answer:**

The hour hand of a clock makes one complete clockwise rotation over the course of 12 hours, or

minutes.

Therefore, over the course of 20 minutes, the hour hand rotates

.

### Example Question #1 : Rotations

Over the course of one minute and forty seconds, the minute hand of a clock undergoes which of the following clockwise rotations?

**Possible Answers:**

**Correct answer:**

The minute hand of a clock rotates one complete clockwise turn about its mount over the course of 60 minutes, or, equivalently, since there are 60 seconds in a minute, every

seconds.

Over the course of 1 minute 40 seconds - or, since one minute is equal to 60 seconds, seconds - the minute hand rotates clockwise

.

### Example Question #1 : Rotations

Examine the figures in the above diagram. The figure at right is the result of performing which of the following transformations on the figure at left?

**Possible Answers:**

A clockwise rotation

A counterclockwise rotation

A counterclockwise rotation

A clockwise rotation

A rotation

**Correct answer:**

A clockwise rotation

Examine the figure below:

If we connect the horizontal line with the line along the rotated nine at right, we see that it is the result of a one-third turn clockwise; the angle between them

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