HiSET: Math : Understand transformations in the plane

Study concepts, example questions & explanations for HiSET: Math

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Example Questions

Example Question #1 : Rotations

Octagon

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:

Explanation:

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

Octagon

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:

Octagon

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

Example Question #1 : Rotations

Hexagon

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:

Explanation:

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

Hexagon

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

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

Hexagon

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:

Explanation:

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:

Explanation:

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:

Explanation:

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

Letters

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:

 counterclockwise rotation

 clockwise rotation

 counterclockwise rotation

 clockwise rotation

 rotation

Correct answer:

 clockwise rotation

Explanation:

Examine the figure below:

1

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 

Example Question #61 : Measurement And Geometry

Letters

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:

 clockwise rotation

 clockwise rotation

 counterclockwise rotation

None of the other choices gives the correct response.

 counterclockwise rotation

Correct answer:

 clockwise rotation

Explanation:

Examine the figure below:

1

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-eighth turn clockwise; the angle between them .

Example Question #11 : Rotations

Letters

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:

 clockwise rotation

 counterclockwise rotation

 counterclockwise rotation

 rotation

 clockwise rotation

Correct answer:

 counterclockwise rotation

Explanation:

Examine the figure below. 

1

If we connect the horizontal line with the line along the rotated "omega" at right, we see that it is the result of a one-sixth turn counterclockwise; the angle between them is one sixth of , or  .

Example Question #13 : Rotations

Over the course of one hour and twenty minutes, the hour hand of a clock undergoes which of the following clockwise rotations?

Possible Answers:

Correct answer:

Explanation:

The hour hand of a clock makes one complete  clockwise rotation over the course of 12 hours, or, since one hour is equal to 60 minutes,

Therefore, over the course of 1 hour 20 minutes - which is equal to minutes - the hour hand rotates

.

Example Question #1 : Dilations

Right triangle

On the above right triangle perform a dilation of scale factor  with the center of the dilation at the circumcenter of the triangle. Let the images of , and  be ,  and , respectively.

Which of the following correctly shows  relative to  ?

Possible Answers:

Right triangle

Right triangle

Right triangle

Right triangle

Right triangle

Correct answer:

Right triangle

Explanation:

The circumcenter of a triangle can be located by finding the intersection of the perpendicular bisectors of the three sides of the triangle. The perpendicular bisectors are shown below, with point of intersection  :

Right triangle

It can be seen that, as is characteristic of a right triangle, this point is the midpoint of the hypotenuse. Construct . A dilation of scale factor   with center  can be performed by letting , and  be the midpoints of , and , respectively: 

Right triangle

Removing the perpendicular bisectors and , we see that the correct choice is the figure

Right triangle

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