body{margin:0;} noscript{z-index: 10000; position: fixed; display: block; height: 100%; width: 100%; background: #fff;} noscript .fa{font-size: 40px; display:block; color: #daa520;} noscript div{margin-top: 6%; padding-left: 20px; padding-right: 20px; text-align: center;} You must have JavaScript enabled to use this site.

Recommended

Tests

An error ocurred, try again later!

eCourses /

Geometry /

Chapter {{ article.chapter.number }}

{{ article.number }}.

	{{ 'ml-lesson-number-slides' \| message : article.intro.bblockCount}}
	{{ 'ml-lesson-number-exercises' \| message : article.intro.exerciseCount}}
	{{ 'ml-lesson-time-estimation' \| message }}

Image Credits

{{ item.file.title }}
{{ presentation }}

No file copyrights entries found

When learning about transformations, rotations were mentioned and used briefly. In this lesson, rotations will be studied more deeply by analyzing the relationships between a point

Q

and its image

Q^{'}

under a rotation.

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Example

Identifying Rotations

Consider the following three triangles and a point

P .

Use the given measuring tool to find the distance from each vertex to

P

and the angles formed by each preimage, the point

P,

and the corresponding image.

Which of the triangles,

△ A^{'} B^{'} C^{'}

△ A^{''} B^{''} C^{''}

, is the image of

△ A B C

under a rotation about

P ?

If one of the triangles is a rotation of

△ A B C

about

P,

what is the measure of the angle of rotation?

Hint

Remember, after performing a rotation, the preimage and the image of a point are the same distance from the center of rotation. The angle of rotation is formed by a preimage, the center of rotation, and the corresponding image.

Solution

Remember that, after performing a rotation, the preimage and the image of a point are the same distance from the center of rotation. Then, start by finding the distances between each vertex and

P .

Notice that the vertices of

△ A^{'} B^{'} C^{'}

are further from

P

than the vertices of

△ A B C .

Consequently,

△ A^{'} B^{'} C^{'}

cannot be the image of

△ A B C

under a rotation about

P .

Next, find and compare the measures of

∠ A P A^{'},

∠ B P B^{'},

and

∠ C P C^{'} .

As can be seen,

∠ A P A^{'},

∠ B P B^{'},

and

∠ C P C^{'}

have all the same measure, which is

15 0^{\circ}

when measured counterclockwise or

21 0^{\circ}

when measured clockwise. Therefore,

△ A^{''} B^{''} C^{''}

is the image of

△ A B C

under a rotation about

P .

The angle of rotation is either

15 0^{\circ}

21 0^{\circ} .

{{ article.displayTitle }}

Catch-Up and Review

Example

Identifying Rotations

Hint

Solution

Discussion

Rotations Performed by Hand