Understand transformations in the plane - HiSET
Card 1 of 164
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 
?
Consider Square 
Which of the following diagrams correctly shows Square 
Tap to reveal answer
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:
Now, do the same thing to the new square, but with 
as the center. The figure is below:
The final image, relative to the original square, is below:
To perform a dilation with center 
Now, do the same thing to the new square, but with 
The final image, relative to the original square, is below:
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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 
?
On the above right triangle perform a dilation of scale factor 
Which of the following correctly shows 
Tap to reveal answer
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:
This figure is the correct choice.
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, 
This is done by mapping 
This figure is the correct choice.
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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 
?
On the above right triangle perform a dilation of scale factor 
Which of the following correctly shows 
Tap to reveal answer
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 
:
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:
Removing the perpendicular bisectors and 
, we see that the correct choice is the figure
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 
It can be seen that, as is characteristic of a right triangle, this point is the midpoint of the hypotenuse. Construct 
Removing the perpendicular bisectors and 
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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 
?
On the above right triangle perform a dilation of scale factor 
Which of the following correctly shows 
Tap to reveal answer
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 
:
A dilation of scale factor 
with center 
can be performed by letting 
, 
, and 
be the midpoints of 
, 
, and 
, respectively:
Removing the medians and 
, we see that the correct choice is the figure
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 
A dilation of scale factor 
Removing the medians and 
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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 
?
Consider Square 
Which of the following diagrams correctly shows Square
Tap to reveal answer
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:
Now, do the same thing to the new square, but with 
as the center. The figure is below:
The image, relative to the original square, is below:
To perform a dilation with center 
Now, do the same thing to the new square, but with 
The image, relative to the original square, is below:
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On the above obtuse 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 
?
On the above obtuse triangle perform a dilation of scale factor
Which of the following correctly shows
Tap to reveal answer
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 
:
Construct 
, 
, and 
. A dilation of scale factor 
with center 
can be performed by letting 
, 
, and 
be the midpoints of 
, 
, and 
, respectively:
Removing the perpendicular bisectors and 
, we see that the correct choice is the figure
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
Construct 
Removing the perpendicular bisectors and
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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 ?
On the above obtuse triangle perform a dilation of scale factor
Which of the following correctly shows
Tap to reveal answer
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 :
A dilation of scale factor with center can be performed by letting , , and be the midpoints of , , and , respectively:
Removing the altitudes and , we see that the correct choice is the figure:
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 
A dilation of scale factor
Removing the altitudes and
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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 ?
On the above obtuse triangle, perform a dilation of scale factor
Which of the following correctly shows
Tap to reveal answer
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 
:
A dilation of scale factor with center can be performed by letting , , and be the midpoints of , , and , respectively:
Removing the medians and , we see that the correct choice is the figure
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
A dilation of scale factor
Removing the medians and
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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.
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.
Tap to reveal answer
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:
Therefore, 
and 
, making the equation of the image
,
or
.
The graph of the equation
is an ellipse whose center is at the origin; it has a horizontal axis with endpoints at 
or
has its center at the origin, and its endpoints at 
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 
or
and the endpoints of the vertical axis will be
or
This dilation is shown in the figure below:
Therefore, 
or
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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.
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.
Tap to reveal answer
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:
Therefore, 
and 
, making the equation of the image
,
or
.
The graph of the equation
is an ellipse whose center is at the origin; it has a horizontal axis with endpoints at 
or
has its center at the origin, and its endpoints at 
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:
Therefore, 
or
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On the graph of the equation
,
perform a dilation with center 
and scale factor 
.
Give the equation of the resulting circle.
On the graph of the equation
perform a dilation with center 
Give the equation of the resulting circle.
Tap to reveal answer
The graph of the equation
is a circle with center 
and radius 
.
,
or
is a circle with center at origin 
and radius 8.
A dilation of a circle with scale factor 
will result in multiplying that radius by 
, so the radius of the circle will be 
.
To find the center of the image, note that the origin is 4 units below 
. The center of the new circle must be 
units below 
, so this center will be 
, or 
. See the figure below:
Substituting in the circle formula, this is
,
or
.
The graph of the equation
is a circle with center 
or
is a circle with center at origin 
A dilation of a circle with scale factor 
To find the center of the image, note that the origin is 4 units below 
Substituting in the circle formula, this is
or
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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?
On the coordinate plane, Quadrilateral 
Perform a dilation of this quadrilateral with center at the origin and scale factor 
Which of the following choices does not match the point with its correct coordinates?
Tap to reveal answer
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 
.
The image of a point 
The images of the other three vertices can be found similarly:
Note that only in the case of 
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What is the result of reflecting the point 
over the y-axis in the coordinate plane?
What is the result of reflecting the point
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Reflecting a point
over the y-axis geometrically is the same as negating the x-coordinate of the ordered pair to obtain
.
Thus, since our initial point was
and we want to reflect it over the y-axis, we obtain the reflection by negating the first term of the ordered pair to get
.
Reflecting a point
over the y-axis geometrically is the same as negating the x-coordinate of the ordered pair to obtain
Thus, since our initial point was
and we want to reflect it over the y-axis, we obtain the reflection by negating the first term of the ordered pair to get
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How many lines of symmetry does the above figure have?
How many lines of symmetry does the above figure have?
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A line of symmetry of a figure is about which the reflection of the figure is the figure itself. The diagram below shows the only line of symmetry of the figure.
The correct response is one.
A line of symmetry of a figure is about which the reflection of the figure is the figure itself. The diagram below shows the only line of symmetry of the figure.
The correct response is one.
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On the coordinate plane, the point 
is reflected about the 
-axis. The image is denoted 
. Give the distance 
.
On the coordinate plane, the point 
Tap to reveal answer
The reflection of the point at 
about the 
-axis is the point at 
; therefore, the image of is 
. Since these two points have the same 
-coordinate, the distance between them is the absolute value of the difference between their 
-coordinates:
The reflection of the point at 
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Consider regular Hexagon 
; let 
and 
be the midpoints of 
and 
. Reflect the hexagon about 
, then again about 
. With which of the following points does the image of 
under these reflections coincide?
Consider regular Hexagon 
Tap to reveal answer
Refer to the figure below, which shows the reflection of 
about 
; we will call this image 
.
Note that 
coincides with 
. Now, refer to the figure below, which shows the reflection of 
about 
; we will call this image - the final image - 
Note that 
coincides with 
, making this the correct response.
Refer to the figure below, which shows the reflection of 
Note that 
Note that 
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Consider regular Hexagon 
; let 
and 
be the midpoints of 
and 
. Reflect the hexagon about 
, then again about 
. Which of the following clockwise rotations about the center would result in each point being its own image under this series of transformations?
Consider regular Hexagon 
Tap to reveal answer
Refer to the figure below, which shows the reflection of the given hexagon about 
; we will call 
the image of 
, call 
the image of 
, and so forth.
Now, refer to the figure below, which shows the reflection of the image about 
; we will call 
the image of 
, call 
the image of 
, and so forth.
Note that the vertices coincide with those of the original hexagon, and that the images of the points are in the same clockwise order as the original points. Since 
coincides with 
, 
coincides with 
, and so forth, a clockwise rotation of five-sixth of a complete turn - that is, 
,
is required to make each point its own image under the three transformations.
Refer to the figure below, which shows the reflection of the given hexagon about 
Now, refer to the figure below, which shows the reflection of the image about 
Note that the vertices coincide with those of the original hexagon, and that the images of the points are in the same clockwise order as the original points. Since 
is required to make each point its own image under the three transformations.
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What is the result of rotating the point 
about the origin in the plane by 
?
What is the result of rotating the point
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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 
.
Rotating a point
geometrically in the plane about the origin
Thus, since our initial point was
we negate both coordinates to get
as the rotation about the origin by
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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?
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?
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Examine the figure below.
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 
.
Examine the figure below.
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 
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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?
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?
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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 
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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