HiSET: Math : Dilations

Study concepts, example questions & explanations for HiSET: Math

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Example Questions

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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

Example Question #2 : Dilations

Right triangle

On the above right triangle perform a dilation of scale factor  with the center of the dilation at the centroid 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 centroid of a triangle can be located by finding the intersection of the three medians of the triangle - the segments that connect each vertex to the midpoint of its opposite side. The medians are shown below, with point of intersection  :

Right triangle

A dilation of scale factor   with center  can be performed by letting , and  be the midpoints of , , and , respectively: 

Right triangle

Removing the medians and , we see that the correct choice is the figure

Right triangle

Example Question #3 : Dilations

On the coordinate plane, Quadrilateral  has its vertices at the following four points:

Perform a dilation of this quadrilateral with center at the origin and scale factor . Call the image Quadrilateral , where  is the image of , etc.

Which of the following choices does not match the point with its correct coordinates?

Possible Answers:

All four of the other choices correctly match their points with their coordinates.

Correct answer:

Explanation:

The image of a point  under a dilation with center  and scale factor  is the point in the same direction from  as , but  units away, where  is the distance from  to . Since the center of the dilation is the origin , the coordinates of the image of , which is , can be found by multiplying scale factor  by each of those of :

The images of the other three vertices can be found similarly:

 

 

 

Note that only in the case of , the ordered pair in the choice differs (in abscissa) from the correct ordered pair. This makes the correct choice .

Example Question #4 : Dilations

Consider Square . Perform two dilations successively, each with scale factor ; the first dilation should have center , the second, . Call the image of  under these dilations ; the image of , and so forth.

Which of the following diagrams correctly shows Square  relative to Square ?  

Possible Answers:

Squares

Squares

Squares

None of the other choices gives the correct response.

Squares

Correct answer:

Squares

Explanation:

To perform a dilation with center  and scale factor , find the midpoints of the segments connecting  to each point, and connect those points. We can simplify the process by finding the midpoints of , and , and naming them , and , respectively; , the image of center , is just  itself. The figure is below:

Square 4

Now, do the same thing to the new square, but with  as the center. The figure is below:

1

The final image, relative to the original square, is below:

Square 4

Example Question #5 : Dilations

Obtuse triangle

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

Which of the following correctly shows  relative to  ?

Possible Answers:

Obtuse triangle

1

Obtuse triangle

Obtuse triangle

Obtuse triangle

Correct answer:

1

Explanation:

The centroid of a triangle can be located by finding the intersection of the three medians of the triangle - the segments that connect each vertex to the midpoint of its opposite side. The medians are shown below, with point of intersection  :

Obtuse triangle

A dilation of scale factor   with center  can be performed by letting , and  be the midpoints of , and , respectively: 

Obtuse triangle

Removing the medians and , we see that the correct choice is the figure

Obtuse triangle

Example Question #6 : Dilations

On the graph of the equation

,

perform a dilation with center at the origin and with scale factor .

Give the equation of the resulting figure.

Possible Answers:

Correct answer:

Explanation:

The graph of the equation

is an ellipse whose center is at the origin; it has a horizontal axis with endpoints at  and a vertical axis with endpoints at  . Thus,

,

or

has its center at the origin, and its endpoints at  and .

The center of dilation is the center of the ellipse, so the center of the image will also be at the origin. The endpoints of the ellipse will be the points  times the distance away from the origin, in the same directions. Since the center of the dilation is the origin, these points can be found by simply multiplying the coordinates by ; the endpoints of the horizontal axis will be

,

or

,

and the endpoints of the vertical axis will be

or

.

This dilation is shown in the figure below:

1

Therefore,  and , making the equation of the image

,

or

.

 

Example Question #7 : Dilations

Right triangle

On the above right triangle perform a dilation of scale factor  with the center of the dilation at the orthocenter 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 orthocenter of a triangle can be located by finding the intersection of the three altitudes of the triangle - the segments connecting each vertex to its opposite side, perpendicular to the respective side. Since the triangle is right,  and  are two of the altitudes, which intersect at ; the third altitude must also pass through , since the three altitudes are concurrent. Therefore, we perform a dilation of the triangle with respect to center .

This is done by mapping  and  to the midpoints of  and , respectively, and by mapping  to itself. The triangle is seen below:

1

This figure is the correct choice.

Example Question #8 : Dilations

Consider Square . Perform two dilations successively, each with scale factor ; the first dilation should have center , the second, . Call the image of  under these dilations ; the image of , and so forth.

Which of the following diagrams correctly shows Square  relative to Square ?  

Possible Answers:

Squares

Squares

Squares

Squares

Squares

Correct answer:

Squares

Explanation:

To perform a dilation with center  and scale factor , find the midpoints of the segments connecting  to each point, and connect those points. We can simplify the process by finding the midpoints of , and , and naming them , and , respectively; , the image of center , is just  itself. The figure is below:

1

Now, do the same thing to the new square, but with  as the center. The figure is below:

1

The image, relative to the original square, is below:

 Squares

Example Question #9 : Dilations

On the graph of the equation

,

perform a dilation with center at the origin and with scale factor 3.

Give the equation of the resulting image.

Possible Answers:

Correct answer:

Explanation:

The graph of the equation

is an ellipse whose center is at the origin; it has a horizontal axis with endpoints at  and a vertical axis with endpoints at  . Thus,

,

or

,

has its center at the origin, and its endpoints at and .

The center of dilation is the center of the ellipse, so the center of the image will also be at the origin. The endpoints of the ellipse will be the points 3 times the distance away from the origin, in the same directions. Since the center of the dilation is the origin, these points can be found by simply multiplying the coordinates by 3; the endpoints of the horizontal axis will be

,

or

,

and the endpoints of the vertical axis will be

or

.

The figure is shown below:

1

Therefore, and , making the equation of the image

,

or

.

Example Question #10 : Dilations

Obtuse triangle

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

Which of the following correctly shows  relative to  ?

Possible Answers:

Obtuse triangle

Obtuse triangle

Obtuse triangle

1

Obtuse triangle

Correct answer:

Obtuse triangle

Explanation:

The orthocenter of a triangle can be located by finding the intersection of the three altitudes of the triangle - the segments connecting each vertex to the line containing its opposite side, perpendicular to the respective side. Extend  and  to  and ;  the altitudes are shown below, with point of intersection  :

Obtuse triangle

A dilation of scale factor   with center  can be performed by letting , and  be the midpoints of , and , respectively: 

Obtuse triangle

Removing the altitudes and , we see that the correct choice is the figure:

Obtuse triangle

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