Common Core: High School - Geometry : Drawing Transformed Figures: CCSS.Math.Content.HSG-CO.A.5

Study concepts, example questions & explanations for Common Core: High School - Geometry

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Example Questions

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Example Question #1 : Draw Generalized Transformed Figures

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?

Possible Answers:

Rotation

Reflection

None of the other answers.

Translation

Extension

Correct answer:

Translation

Explanation:

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.

Example Question #2 : Draw Generalized Transformed Figures

If a rectangle has the coordinate values, , , , and  and after a transformation results in the coordinates , , , and  identify the transformation.

Possible Answers:

None of the other answers.

Extension

Dilation

Reflection

Rotation

Correct answer:

Reflection

Explanation:

"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.

Example Question #3 : Draw Generalized Transformed Figures

If a rectangle has the coordinate values, , and  and after a transformation results in the coordinates , and  identify the transformation.

Possible Answers:

Rotation

Dilation

None of the other answers.

Translation

Reflection

Correct answer:

Translation

Explanation:

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.

Example Question #4 : Draw Generalized Transformed Figures

If a rectangle has the coordinate values, , and  and after a transformation results in the coordinates , and  identify the transformation.

Possible Answers:

Reflection

Dilation

Rotations

Extension

None of the others

Correct answer:

Reflection

Explanation:

"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.

Example Question #1 : Draw Generalized Transformed Figures

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.

Possible Answers:

Rotation

Transformation

Dilation

None of the above

Reflection

Correct answer:

Dilation

Explanation:

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. 

Screen shot 2020 07 06 at 4.40.23 pm

Example Question #2 : Draw Generalized Transformed Figures

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?

Possible Answers:

(a,b), (c,d), and (e,f)

(-a,b), (-c,d), and (-e,f)

(a,b), (-c,-d), and (-e,-f)

(-a,-b), (-c,-d), and (-e,-f)

(a,-b), (c,-d), and (e,-f)

Correct answer:

(-a,-b), (-c,-d), and (-e,-f)

Explanation:

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. 

Screen shot 2020 07 06 at 4.54.11 pm

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). 

Example Question #1 : Draw Specific Transformed Figures

The coordinates of a trapezoid are, , , , and . What are the coordinates of this trapezoid after it is reflected across the -axis?

Possible Answers:

Correct answer:

Explanation:

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.

Screen shot 2016 06 15 at 5.49.55 am

From here, to reflect the image across the -axis take the negative of all the  values.

This change results in the following,

Screen shot 2016 06 15 at 5.51.13 am

Therefore, the coordinates of the reflected trapezoid are

Example Question #2 : Draw Specific Transformed Figures

The coordinates of a trapezoid are, , , , and . What are the coordinates of this trapezoid after it is reflected across the -axis?

Possible Answers:

Correct answer:

Explanation:

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.

Screen shot 2016 06 15 at 5.49.55 am

From here, to reflect the image across the -axis take the opposite of all the  values.

This change results in the following,

Screen shot 2016 06 15 at 6.02.01 am

Therefore, the coordinates of the reflected trapezoid are

Example Question #571 : High School: Geometry

The coordinates of a trapezoid are, , , , and . What are the coordinates of this trapezoid after it is reflected across the  and -axis?

Possible Answers:

Correct answer:

Explanation:

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.

Screen shot 2016 06 15 at 5.49.55 am

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,

Screen shot 2016 06 15 at 6.19.15 am

Therefore, the coordinates of the reflected trapezoid are

Example Question #4 : Draw Specific Transformed Figures

The coordinates of a triangle are, , , and . What are the coordinates of this triangle after it is reflected across the -axis?

Possible Answers:

Correct answer:

Explanation:

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.

Screen shot 2016 06 15 at 6.33.25 am

From here, to reflect the image across the -axis take the negative of all the  values.

This change results in the following,

Screen shot 2016 06 15 at 6.40.50 am

Therefore, the reflected triangle has coordinates at

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