Common Core: High School - Geometry : Congruence

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

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All Common Core: High School - Geometry Resources

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Example Questions

Example Question #4 : Rotations And Reflections Of Rectangles, Parallelograms, Trapezoids, And Regular Polygons: Ccss.Math.Content.Hsg Co.A.3

If the rectangle is rotated clockwise around the origin  degrees, what is the resulting image?

Rotation

Possible Answers:

Rotation3

Rotation

Rotation3

None of the answers.

Rotation3

Correct answer:

Rotation3

Explanation:

To rotate the rectangular object around the origin, first recall the definition for a rotation and origin.

Rotation: To rotate an object either clockwise or counter clockwise around a center point. In this particular case the center point is the origin or the point  where the x and y axis intersect.

Looking at the original image and making one 90 degree rotation around the origin results in the following.

Rotation

Rotation1

When rotating, the bottom right point will become the bottom left point, the top right point becomes the bottom right point, the left bottom point becomes the top left point, and the left top point becomes the top right point. 

Now, rotating it another  degrees to get to the  degree rotated image, results in the following

Rotation3

Example Question #5 : Rotations And Reflections Of Rectangles, Parallelograms, Trapezoids, And Regular Polygons: Ccss.Math.Content.Hsg Co.A.3

If the rectangle is rotated clockwise around the origin  degrees, what is the resulting image?

Rotation

Possible Answers:

Rotation3

Rotation3

Rotation3

Rotation

None of the answers.

Correct answer:

Rotation3

Explanation:

To rotate the rectangular object around the origin, first recall the definition for a rotation and origin.

Rotation: To rotate an object either clockwise or counter clockwise around a center point. In this particular case the center point is the origin or the point  where the x and y axis intersect.

Looking at the original image and making one rotation around the origin results in the following.

Rotation

Rotation1

When rotating, the bottom right point will become the bottom left point, the top right point becomes the bottom right point, the left bottom point becomes the top left point, and the left top point becomes the top right point. The visual representation for this rotation is as follows.

Rotation3

Example Question #6 : Rotations And Reflections Of Rectangles, Parallelograms, Trapezoids, And Regular Polygons: Ccss.Math.Content.Hsg Co.A.3

If the rectangle is rotated clockwise around the origin  degrees, what is the resulting image?

Rotation

Possible Answers:


Rotation3

Rotation3

Rotation3

Rotation3

Rotation

Correct answer:


Rotation3

Explanation:

To rotate the rectangular object around the origin, first recall the definition for a rotation and origin.

Rotation: To rotate an object either clockwise or counter clockwise around a center point. In this particular case the center point is the origin or the point  where the x and y axis intersect.

Looking at the original image and making one  degree rotation around the origin results in the following.

Rotation

Rotation1

When rotating, the bottom right point will become the bottom left point, the top right point becomes the bottom right point, the left bottom point becomes the top left point, and the left top point becomes the top right point. 

Now, rotating it another  degrees to get to the  degree rotated image, results in the following

Rotation3

From here one final rotation must occur to reach  degrees.

Rotation3

Example Question #7 : Rotations And Reflections Of Rectangles, Parallelograms, Trapezoids, And Regular Polygons: Ccss.Math.Content.Hsg Co.A.3

If the rectangle is rotated clockwise around the origin  degrees, what is the resulting image?

Rotation

Possible Answers:

Rotation

Rotation3

Rotation3

Rotation3

Rotation3

Correct answer:

Rotation

Explanation:

To rotate the rectangular object around the origin, first recall the definition for a rotation and origin.

Rotation: To rotate an object either clockwise or counter clockwise around a center point. In this particular case the center point is the origin or the point  where the x and y axis intersect.

Looking at the original image and making one 90 degree rotation around the origin results in the following.

Rotation

Rotation1

When rotating, the bottom right point will become the bottom left point, the top right point becomes the bottom right point, the left bottom point becomes the top left point, and the left top point becomes the top right point. 

Now, rotating it another  degrees to get to the  degree rotated image, results in the following

Rotation3

From here another rotation must occur to reach  degrees.

Rotation3

Lastly, complete one more  degree rotation to land at the  degree mark. Notice that rotating 360 degrees lands the image back at its original spot.

Rotation

Example Question #31 : High School: Geometry

Determine whether the statement is true or false:

An equilateral diamond when rotated  degrees around its center results in the reflected image of two isosceles triangle.

Possible Answers:

True

False

Correct answer:

True

Explanation:

To determine whether this statement is true or false first recall what a  rotation and what a reflection is.

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.

Now looking at the statement, draw images that depict the 90 degree rotation and the reflection of the original image to verify the truth of the statement.

"An equilateral diamond when rotated 90 degrees around its center results in the reflected image of two isosceles triangle."

 

The original equilateral diamond is:

Screen shot 2016 06 14 at 10.13.14 am

The diamond rotated  degrees:

Screen shot 2016 06 14 at 10.16.19 am

The reflection of an isosceles triangle.

Screen shot 2016 06 14 at 10.20.12 am

Therefore, the statement is true.

Example Question #32 : High School: Geometry

Determine whether the statement is true or false:

An equilateral diamond when rotated  degrees around its right most corner results in the reflected image of the original diamond.

Possible Answers:

False

True

Correct answer:

True

Explanation:

To determine whether this statement is true or false first recall what a  rotation and what a reflection is.

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.

Now looking at the statement, draw images that depict the  degree rotation and the reflection of the original image to verify the truth of the statement.

"An equilateral diamond when rotated  degrees around its right most corner results in the reflected image of the original diamond."

The original equilateral diamond with rotation of  degrees on its right most corner is as follows:

Screen shot 2016 06 14 at 10.16.19 amScreen shot 2016 06 14 at 11.02.30 am

The rotated image results in the reflected original image.

Therefore, the statement is true. 

Example Question #1 : Rotation, Reflection, And Transformation Definitions: Ccss.Math.Content.Hsg Co.A.4

Two pencils are placed equidistance away from one another on a table. If these two pencils are parallel, what is the result after they are reflected.

Possible Answers:

The two pencils are now perpendicular.

The two pencils are still parallel.

None of the answers.

The two pencils do not move.

The two pencils are located on the same line.

Correct answer:

The two pencils are still parallel.

Explanation:

Before solving this problem recall the definition of a reflection.

Reflection: The transformation of each and every point on a line or figure across some line  where  is some constant or linear function.

Applying this definition to the situation at hand result in the following depiction. The pencils are represented by the blue and black lines and the reflection line is the red dashed line. After the reflection occurs, the pencils essentially swap positions. Since the pencils were parallel to start with, they remain parallel after the reflection occurred.

Pencil reflection

Therefore, the correct answer is,

"The two pencils are still parallel."

Example Question #2 : Rotation, Reflection, And Transformation Definitions: Ccss.Math.Content.Hsg Co.A.4

Given a circle that is divided into  equal pieces, what is the number of rotations that can occur to keep symmetry?

Possible Answers:

Twelve

Three

None of the answers.

Five

Six

Correct answer:

Six

Explanation:

Drawing the described circle is the first step in answering this given question.

Circle6pieces

To calculate the number of symmetrical rotations, take one piece and rotate it clockwise until it results in the exact image as originally started with. Since the circle is divided into six equal pieces that means that rotating one of the pieces can be done six different times and still keep rotational symmetry, which also means keeping the angles of each piece the same and returning to the original image. 

Therefore the correct answer is six.

Example Question #31 : Congruence

Given a regular polygon that has  sides, calculate the order of rotational symmetry.

Possible Answers:

Correct answer:

Explanation:

For this particular question recall what rotational symmetry means. For an object to achieve rotational symmetry it must return to its original form. This can be accomplished by rotating the figure or equal parts of the figure in either the clockwise or non clockwise direction. 

For this particular problem, it is stated that the polygon is "regular" meaning all sides and angles are equal to one another. It is also said that the polygon has  sides. This means that the figure can be rotated  times to return to its original form.

Therefore, to obtain rotational symmetry of a regular polygon, it must have an order equal to the number of sides.

Thus, the answer is .

Example Question #31 : Congruence

Inscribed polygon

Looking at the above image, calculate the order of rotational symmetry of the polygon that is inscribed in the circle.

Possible Answers:

Correct answer:

Explanation:

For this particular question recall what rotational symmetry means. For an object to achieve rotational symmetry it must return to its original form. This can be accomplished by rotating the figure or equal parts of the figure in either the clockwise or non clockwise direction. 

For this particular problem, look at the following image. 

Inscribed polygon

The polygon has six sides. This means that the figure can be rotated  times to return to its original form.

Therefore, to obtain rotational symmetry of a regular polygon, it must have an order equal to the number of sides.

Thus, the answer is .

All Common Core: High School - Geometry Resources

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