Drawing Transformed Figures: CCSS.Math.Content.HSG-CO.A.5 - Common Core: High School - Geometry
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If a triangle is in quadrant three and undergoes a transformation that moves each of its coordinate points to the left three units and down one unit, what transformation has occurred?
If a triangle is in quadrant three and undergoes a transformation that moves each of its coordinate points to the left three units and down one unit, what transformation has occurred?
To determine the type of transformation that is occurring in this particular situation, first recall the different types of transformations.
Translation: To move an object from its original position a certain distance without changing the object in any other way.
Transformation: Refers to any of the four changes that can be done to an object geographically. Transformations include, reflections, translations, rotations, and resizing the object.
Reflection: To flip the orientation of an object over a specific line or function.
Rotation: To rotate an object either clockwise or counter clockwise around a center point.
Since each of the triangle's coordinates is moved to the left and down, it is seen that the size and shape of the triangle remains the same but its location is different. Therefore, the transformation the triangle has undergone is a translation.
To determine the type of transformation that is occurring in this particular situation, first recall the different types of transformations.
Translation: To move an object from its original position a certain distance without changing the object in any other way.
Transformation: Refers to any of the four changes that can be done to an object geographically. Transformations include, reflections, translations, rotations, and resizing the object.
Reflection: To flip the orientation of an object over a specific line or function.
Rotation: To rotate an object either clockwise or counter clockwise around a center point.
Since each of the triangle's coordinates is moved to the left and down, it is seen that the size and shape of the triangle remains the same but its location is different. Therefore, the transformation the triangle has undergone is a translation.
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If a rectangle has the coordinate values, 
, 
, 
, and 
and after a transformation results in the coordinates 
, 
, 
, and 
identify the transformation.
If a rectangle has the coordinate values,
"If a rectangle has the coordinate values, 
, 
, 
, and 
and after a transformation results in the coordinates 
, 
, 
, and 
identify the transformation."
A transformation that changes the 
values by multiplying them by negative one is known as a reflection across the 
-axis or the line 
Therefore, this particular rectangle is being reflected across the 
-axis because the opposite of all the 
values have been taken.
"If a rectangle has the coordinate values,
A transformation that changes the
Therefore, this particular rectangle is being reflected across the
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If a rectangle has the coordinate values, 
, 
, 
, and 
and after a transformation results in the coordinates 
, 
, 
, and 
identify the transformation.
If a rectangle has the coordinate values,
"If a rectangle has the coordinate values, 
, 
, 
, and 
and after a transformation results in the coordinates 
, 
, 
, and 
identify the transformation."
A transformation that changes the 
values by multiplying them by negative one is known as a reflection across the 
-axis or the line 
Therefore, this particular rectangle is being reflected across the 
-axis because the opposite of all the 
values have been taken.
"If a rectangle has the coordinate values,
A transformation that changes the
Therefore, this particular rectangle is being reflected across the
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If a rectangle has the coordinate values, 
, 
, 
, and 
and after a transformation results in the coordinates 
, 
, 
, and 
identify the transformation.
If a rectangle has the coordinate values,
To identify the transformation that is occurring in this particular problem, recall the different transformations.
Translation: To move an object from its original position a certain distance without changing the object in any other way.
Transformation: Refers to any of the four changes that can be done to an object geographically. Transformations include, reflections, translations, rotations, and resizing the object.
Reflection: To flip the orientation of an object over a specific line or function.
Rotation: To rotate an object either clockwise or counter clockwise around a center point.
Looking at the starting and ending coordinates of the rectangle,
, 
, 
, and 
to 
, 
, 
, and 
Since all the 
coordinates are increasing by two this is known as a translation.
To identify the transformation that is occurring in this particular problem, recall the different transformations.
Translation: To move an object from its original position a certain distance without changing the object in any other way.
Transformation: Refers to any of the four changes that can be done to an object geographically. Transformations include, reflections, translations, rotations, and resizing the object.
Reflection: To flip the orientation of an object over a specific line or function.
Rotation: To rotate an object either clockwise or counter clockwise around a center point.
Looking at the starting and ending coordinates of the rectangle,
Since all the
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Assume the values a, b, c, and d are all positive integers. If a rectangle has the coordinate values, (a,b), (c,b), (a,d), and (c,d) and after a transformation results in the coordinates (a,b), (2c,b), (a,2d), and (2c,2d) identify the transformation.
Assume the values a, b, c, and d are all positive integers. If a rectangle has the coordinate values, (a,b), (c,b), (a,d), and (c,d) and after a transformation results in the coordinates (a,b), (2c,b), (a,2d), and (2c,2d) identify the transformation.
The above described transformation is a dilation. Notice that one point, (a,b), stays the same before and after the transformation. The point (c,b) retains the same y value of b, but c is dilated into 2c, extending the base of the rectangle. The point (a,d) is similar in that the x value of a stays the same, but the y value of d is extended or dilated to 2d. The final point (c,d) is extended in both length and width to become (2c,2d). The below graph shows the original figure in blue and the dilated larger figure in pink.
The above described transformation is a dilation. Notice that one point, (a,b), stays the same before and after the transformation. The point (c,b) retains the same y value of b, but c is dilated into 2c, extending the base of the rectangle. The point (a,d) is similar in that the x value of a stays the same, but the y value of d is extended or dilated to 2d. The final point (c,d) is extended in both length and width to become (2c,2d). The below graph shows the original figure in blue and the dilated larger figure in pink.
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Imagine a triangle with vertices located at the points (a,b), (c,d), and (e,f). If this figure were rotated 180o about the origin, what would be the new coordinates of the triangle's vertices?
Imagine a triangle with vertices located at the points (a,b), (c,d), and (e,f). If this figure were rotated 180o about the origin, what would be the new coordinates of the triangle's vertices?
The correct answer is (-a,-b), (-c,-d), and (-e,-f). In other words, you'd just take the opposite value of each x and y value of each vertex of the triangle. The following diagram shows one set of vertexes rotated 180o about the origin to help demonstrate this.
Please note that if one of our original points had any negative values, such as the point (2,-2), and we rotated it 180o about the origin, the signs of both the x and y values would change, and this point's image after translation would be (-2,2).
The correct answer is (-a,-b), (-c,-d), and (-e,-f). In other words, you'd just take the opposite value of each x and y value of each vertex of the triangle. The following diagram shows one set of vertexes rotated 180o about the origin to help demonstrate this.
Please note that if one of our original points had any negative values, such as the point (2,-2), and we rotated it 180o about the origin, the signs of both the x and y values would change, and this point's image after translation would be (-2,2).
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The coordinates of a trapezoid are, 
, 
, 
, and 
. What are the coordinates of this trapezoid after it is reflected across the 
-axis?
The coordinates of a trapezoid are, 
To find the reflected image of the trapezoid, first identify how it is being reflected. This particular problem states that it is being reflected over the 
-axis. Recall that the 
-axis is the horizontal axis on the coordinate grid and is equivalent to the line 
.
Plot the points of the original trapezoid on the coordinate grid.
From here, to reflect the image across the 
-axis take the negative of all the 
values.
This change results in the following,
Therefore, the coordinates of the reflected trapezoid are
To find the reflected image of the trapezoid, first identify how it is being reflected. This particular problem states that it is being reflected over the
Plot the points of the original trapezoid on the coordinate grid.
From here, to reflect the image across the
This change results in the following,
Therefore, the coordinates of the reflected trapezoid are
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The coordinates of a trapezoid are, 
, 
, 
, and 
. What are the coordinates of this trapezoid after it is reflected across the 
-axis?
The coordinates of a trapezoid are, 
To find the reflected image of the trapezoid, first identify how it is being reflected. This particular problem states that it is being reflected over the 
-axis. Recall that the 
-axis is the vertical axis on the coordinate grid and is equivalent to the line 
.
Plot the points of the original trapezoid on the coordinate grid.
From here, to reflect the image across the -axis take the opposite of all the values.
This change results in the following,
Therefore, the coordinates of the reflected trapezoid are
To find the reflected image of the trapezoid, first identify how it is being reflected. This particular problem states that it is being reflected over the
Plot the points of the original trapezoid on the coordinate grid.
From here, to reflect the image across the
This change results in the following,
Therefore, the coordinates of the reflected trapezoid are
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The coordinates of a trapezoid are, 
, 
, 
, and 
. What are the coordinates of this trapezoid after it is reflected across the 
and 
-axis?
The coordinates of a trapezoid are, 
To find the reflected image of the trapezoid, first identify how it is being reflected. This particular problem states that it is being reflected over the 
and 
-axis. This means the reflected image will be in the fourth quadrant.
Plot the points of the original trapezoid on the coordinate grid.
From here, to reflect the image across the line both axis take the opposite of all the coordinate values.
This change results in the following,
Therefore, the coordinates of the reflected trapezoid are
To find the reflected image of the trapezoid, first identify how it is being reflected. This particular problem states that it is being reflected over the
Plot the points of the original trapezoid on the coordinate grid.
From here, to reflect the image across the line both axis take the opposite of all the coordinate values.
This change results in the following,
Therefore, the coordinates of the reflected trapezoid are
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The coordinates of a triangle are, 
, 
, and 
. What are the coordinates of this triangle after it is reflected across the -axis?
The coordinates of a triangle are, 
To find the reflected image of the triangle, first identify how it is being reflected. This particular problem states that it is being reflected over the -axis. Recall that the -axis is the horizontal axis on the coordinate grid and is equivalent to the line .
Plot the points of the original triangle on the coordinate grid.
From here, to reflect the image across the -axis take the negative of all the values.
This change results in the following,
Therefore, the reflected triangle has coordinates at
To find the reflected image of the triangle, first identify how it is being reflected. This particular problem states that it is being reflected over the
Plot the points of the original triangle on the coordinate grid.
From here, to reflect the image across the
This change results in the following,
Therefore, the reflected triangle has coordinates at
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The coordinates of a triangle are, 
, 
, and 
. What are the coordinates of this triangle after it is reflected across the 
-axis?
The coordinates of a triangle are, 
To find the reflected image of the triangle, first identify how it is being reflected. This particular problem states that it is being reflected over the 
-axis. Recall that the 
-axis is the vertical axis on the coordinate grid and is equivalent to the line 
.
Plot the points of the original triangle on the coordinate grid.
From here, to reflect the image across the 
-axis take the negative of all the 
values.
This change results in the following,
Therefore, the reflected triangle has coordinates at
To find the reflected image of the triangle, first identify how it is being reflected. This particular problem states that it is being reflected over the
Plot the points of the original triangle on the coordinate grid.
From here, to reflect the image across the
This change results in the following,
Therefore, the reflected triangle has coordinates at
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The coordinates of a triangle are 
, 
, and 
. What are the coordinates of this triangle after it is reflected across the and -axis?
The coordinates of a triangle are 
To find the reflected image of the triangle, first identify how it is being reflected. This particular problem states that it is being reflected over the and -axis. This means the reflected image will be in the fourth quadrant.
Plot the points of the original triangle on the coordinate grid.
From here, to reflect the image across the line both axis take the opposite of all the coordinate values.
This change results in the following,
Therefore, the coordinates of the reflected triangle are
To find the reflected image of the triangle, first identify how it is being reflected. This particular problem states that it is being reflected over the
Plot the points of the original triangle on the coordinate grid.
From here, to reflect the image across the line both axis take the opposite of all the coordinate values.
This change results in the following,
Therefore, the coordinates of the reflected triangle are
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A rectangle's coordinate points are 
, 
, 
, and 
. If the rectangle is translated down seven units what are the coordinates of the translated rectangle?
A rectangle's coordinate points are 
To find the coordinates of the translated rectangle, first recall what a translation is. A translation is a shift of the original object without changing the shape of size of the object. In this particular case the starting coordinates of the rectangle are given and the goal is to move the rectangle down seven units. A shift down means a algebraic change in the 
coordinate.
The original rectangle is
If each point on the rectangle is shifted down seven units it results in the following
Therefore, the coordinate points of the translated rectangle are
These coordinates can also be found algebraically by subtracting seven from each 
value.
To find the coordinates of the translated rectangle, first recall what a translation is. A translation is a shift of the original object without changing the shape of size of the object. In this particular case the starting coordinates of the rectangle are given and the goal is to move the rectangle down seven units. A shift down means a algebraic change in the
The original rectangle is
If each point on the rectangle is shifted down seven units it results in the following
Therefore, the coordinate points of the translated rectangle are
These coordinates can also be found algebraically by subtracting seven from each
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A rectangle's coordinate points are 
, 
, 
, and 
. If the rectangle is translated up 
units and to the right 
units what are the coordinates of the translated rectangle?
A rectangle's coordinate points are 
To find the coordinates of the translated rectangle, first recall what a translation is. A translation is a shift of the original object without changing the shape of size of the object. In this particular case the starting coordinates of the rectangle are given and the goal is to move the rectangle up two units and to the right 5 units. A shift up means a algebraic change in the 
coordinate and a shift to the right means an algebraic change to the 
coordinate.
The original rectangle is
If each point on the rectangle is shifted up two units and to the right five units results in the following
Therefore, the coordinate points of the translated rectangle are
To find the coordinates of the translated rectangle, first recall what a translation is. A translation is a shift of the original object without changing the shape of size of the object. In this particular case the starting coordinates of the rectangle are given and the goal is to move the rectangle up two units and to the right 5 units. A shift up means a algebraic change in the
The original rectangle is
If each point on the rectangle is shifted up two units and to the right five units results in the following
Therefore, the coordinate points of the translated rectangle are
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