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1.5: Rotations and Reflections of Angles - Mathematics


Now that we know how to deal with angles of any measure, we will take a look at how certain geometric operations can help simplify the use of trigonometric functions of any angle, and how some basic relations between those functions can be made. The two operations on which we will concentrate in this section are rotation and reflection.

To rotate an angle means to rotate its terminal side around the origin when the angle is in standard position. For example, suppose we rotate an angle ( heta ) around the origin by (90^circ ) in the counterclockwise direction. In Figure 1.5.1 we see an angle ( heta ) in QI which is rotated by (90^circ ), resulting in the angle ( heta + 90^circ ) in QII. Notice that the complement of ( heta ) in the right triangle in QI is the same as the supplement of the angle ( heta + 90^circ ) in QII, since the sum of ( heta ), its complement, and (90^circ ) equals (180^circ ). This forces the other angle of the right triangle in QII to be ( heta ).

Thus, the right triangle in QI is similar to the right triangle in QII, since the triangles have the same angles. The rotation of ( heta ) by (90^circ ) does not change the length (r ) of its terminal side, so the hypotenuses of the similar right triangles are equal, and hence by similarity the remaining corresponding sides are also equal. Using Figure 1.5.1 to match up those corresponding sides shows that the point ((-y,x) ) is on the terminal side of ( heta + 90^circ) when ((x,y) ) is on the terminal side of ( heta ). Hence, by definition,

[ onumber sin;( heta + 90^circ) ~ = ~ frac{x}{r} ~=~ cos; heta ~,~~
cos;( heta + 90^circ) ~ = ~ frac{-y}{r} ~=~ -sin; heta ~,~~
an;( heta + 90^circ) ~ = ~ frac{x}{-y} ~=~ -cot; heta ~.]
Though we showed this for ( heta ) in QI, it is easy (see Exercise 4) to use similar arguments for the other quadrants. In general, the following relations hold for all angles ( heta):

[sin;( heta + 90^circ) ~ = ~ cos; heta label{1.4}]

[ cos;( heta + 90^circ) ~ = ~ -sin; heta label{1.5}]

[ an;( heta + 90^circ) ~ = ~ -cot; heta label{1.6}]

Example 1.26

Recall that any nonvertical line in the (xy)-coordinate plane can be written as (y=mx+b ), where (m ) is the slope of the line (defined as (m = frac{ ext{rise}}{ ext{run}} ) ) and (b ) is the (y)-intercept}, i.e. where the line crosses the (y)-axis (see Figure 1.5.2(a)). We will show that the slopes of perpendicular lines are negative reciprocals. That is, if (y=m_{1}x+b_1 ) and (y=m_{2}x+b_2 ) are nonvertical and nonhorizontal perpendicular lines, then (m_2 = -frac{1}{m_1} ) (see Figure 1.5.2(b)).

First, suppose that a line (y=mx+b ) has nonzero slope. The line crosses the (x)-axis somewhere, so let ( heta ) be the angle that the positive (x)-axis makes with the part of the line above the (x)-axis, as in Figure 1.5.3. For (m > 0 ) we see that ( heta ) is acute and ( an; heta = frac{ ext{rise}}{ ext{run}} = m ).

If (m < 0 ), then we see that ( heta ) is obtuse and the rise is negative. Since the run is always positive, our definition of ( an; heta ) from Section 1.4 means that ( an; heta = frac{- ext{rise}}{- ext{run}} = frac{ ext{rise}}{ ext{run}} = m ) (just imagine in Figure 1.5.3(b) the entire line being shifted horizontally to go through the origin, so that ( heta ) is unchanged and the point ((- ext{run},- ext{rise}) ) is on the terminal side of ( heta)). Hence:

For a line (y=mx+b ) with (m e 0 ), the slope is given by (m = an; heta, ), where ( heta ) is the angle formed by the positive (x)-axis and the part of the line above the (x)-axis.}

Now, in Figure 1.5.2(b) we see that if two lines (y=m_{1}x+b_1 ) and (y=m_{2}x+b_2 ) are perpendicular then rotating one line counterclockwise by (90^circ ) around the point of intersection gives us the second line. So if ( heta ) is the angle that the line (y=m_{1}x+b_1 ) makes with the positive (x)-axis, then ( heta + 90^circ ) is the angle that the line (y=m_{2}x+b_2 ) makes with the positive (x)-axis. So by what we just showed, (m_1 = an; heta ) and (m_2 = an;( heta + 90^circ) ). But by formula Equation ef{1.6} we know that ( an;( heta + 90^circ) = -cot; heta ). Hence, (m_2 = -cot; heta = -frac{1}{ an; heta} = -frac{1}{m_1} ). ( extbf{QED})

Rotating an angle ( heta ) by (90^circ ) in the clockwise direction results in the angle ( heta - 90^circ ). We could use another geometric argument to derive trigonometric relations involving ( heta - 90^circ ), but it is easier to use a simple trick: since Equations ef{1.4}- ef{1.6} hold for any angle ( heta ), just replace ( heta ) by ( heta - 90^circ ) in each formula. Since (( heta - 90^circ) + 90^circ = heta ), this gives us:

[label{1.7} sin;( heta - 90^circ) ~ = ~ -cos; heta]

[label{1.8} cos;( heta - 90^circ) ~ = ~ sin; heta]

[label{1.9} an;( heta - 90^circ) ~ = ~ -cot; heta]

We now consider rotating an angle ( heta ) by (180^circ ). Notice from Figure 1.5.4 that the angles ( heta pm 180^circ ) have the same terminal side, and are in the quadrant opposite ( heta ).

Since ((-x,-y) ) is on the terminal side of ( heta pm 180^circ ) when ((x,y) ) is on the terminal side of ( heta ), we get the following relations, which hold for all ( heta):

[label{1.10}sin;( heta pm 180^circ) ~ = ~ -sin; heta]

[label{1.11} cos;( heta pm 180^circ) ~ = ~ -cos; heta]

[label{1.12} an;( heta pm 180^circ) ~ = ~ an; heta]

A reflection is simply the mirror image of an object. For example, in Figure 1.5.5 the original object is in QI, its reflection around the (y)-axis is in QII, and its reflection around the (x)-axis is in QIV. Notice that if we first reflect the object in QI around the (y)-axis and then follow that with a reflection around the (x)-axis, we get an image in QIII. That image is the reflection around the origin of the original object, and it is equivalent to a rotation of (180^circ ) around the origin. Notice also that a reflection around the (y)-axis is equivalent to a reflection around the (x)-axis followed by a rotation of (180^circ ) around the origin.

Applying this to angles, we see that the reflection of an angle ( heta ) around the (x)-axis is the angle (- heta ), as in Figure 1.5.6.

So we see that reflecting a point ((x,y) ) around the (x)-axis just replaces (y ) by (-y ). Hence:

[label{1.13}sin;(- heta) ~ = ~ -sin; heta]

[label{1.14} cos;(- heta) ~ = ~ cos; heta]

[label{1.15} an;(- heta) ~ = ~ - an; heta]

Notice that the cosine function does not change in Equation ef{1.14} because it depends on (x ), and not on (y ), for a point ((x,y) ) on the terminal side of ( heta ).

In general, a function (f(x) ) is an even function if (f(-x) = f(x)) for all (x ), and it is called an odd function if (f(-x) = -f(x) ) for all (x ). Thus, the cosine function is even, while the sine and tangent functions are odd.

Replacing ( heta ) by (- heta ) in Equations ef{1.4}- ef{1.6}, then using Equations ef{1.13}- ef{1.15}, gives:

[label{1.16}sin;(90^circ - heta) ~ = ~ cos; heta]

[label{1.17}cos;(90^circ - heta) ~ = ~ sin; heta]

[label{1.18} an;(90^circ - heta) ~ = ~ cot; heta]

Note that Equations ef{1.16}- ef{1.18} extend the Cofunction Theorem from Section 1.2 to all ( heta ), not just acute angles. Similarly, Equations ef{1.10}- ef{1.12} and ef{1.13}- ef{1.15} give:

[label{1.19} sin;(180^circ - heta) ~ = ~ sin; heta]

[label{1.20}cos;(180^circ - heta) ~ = ~ -cos; heta]

[label{1.21} an;(180^circ - heta) ~ = ~ - an; heta]

Notice that reflection around the (y)-axis is equivalent to reflection around the (x)-axis (( heta mapsto - heta)) followed by a rotation of (180^circ ) ((- heta mapsto - heta + 180^circ = 180^circ - heta)), as in Figure 1.5.7.

It may seem that these geometrical operations and formulas are not necessary for evaluating the trigonometric functions, since we could just use a calculator. However, there are two reasons for why they are useful. First, the formulas work for any angles, so they are often used to prove general formulas in mathematics and other fields, as we will see later in the text. Second, they can help in determining which angles have a given trigonometric function value.

Example 1.27

Find all angles (0^circ le heta < 360^circ ) such that (sin; heta = -0.682 ).

Solution

Using the (fbox{(sin^{-1})}) button on a calculator with (-0.682 ) as the input, we get ( heta = -43^circ ), which is not between (0^circ ) and (360^circ ). Since ( heta = -43^circ ) is in QIV, its reflection (180^circ - heta ) around the (y)-axis will be in QIII and have the same sine value. But (180^circ - heta = 180^circ - (-43^circ) = 223^circ ) (see Figure 1.5.8). Also, we know that (-43^circ ) and (-43^circ + 360^circ = 317^circ ) have the same trigonometric function values. So since angles in QI and QII have positive sine values, we see that the only angles between (0^circ ) and (360^circ ) with a sine of (-0.682 ) are (oxed{ heta = 223^circ ~ ext{and}~ 317^circ}~ ).


Reflections, Rotations, and Translations

In this task, using computer software, you will apply reflections, rotations, and translations to a triangle. You will then study what happens to the side lengths and angle measures of the triangle after these transformations have been applied. In each part of the question, a sample picture of the triangle is supplied along with a line of reflection, angle of rotation, and segment of translation: the attached GeoGebra software will allow you to experiment with changing the location of the line of reflection, changing the measure of the angle of rotation, and changing the location and length of the segment of translation.

Below is a triangle $ABC$ and a line $overleftrightarrow$:

Use the supplied GeoGebra application to reflect $ riangle ABC$ over $overleftrightarrow$. Label the reflected triangle $A'B'C'$. What are the side lengths and angle measures of triangle $A'B'C'$? What happens when you change the location of one of the vertices of $ riangle ABC$? What happens when you change the location of line $overleftrightarrow$?

Below is a triangle $ABC$ and a point $E$. Draw the rotation of $ riangle ABC$ about $E$ through an angle of 85 degrees in the counterclockwise direction.

Label the image of $ riangle ABC$ as $ riangle A'B'C'$. What happens to the side lengths and angle measures of $ riangle A'B'C'$ when you change the measure of the angle of rotation? What happens when you move the center of rotation $E$?

Below is a triangle $ABC$ and a directed line segment $overline$.

Draw the translation of $ riangle ABC$ by $overline$ and label it $ riangle A'B'C'$. What happens to the side lengths and angle measures of triangle $A'B'C'$ when you change one of the vertices, $A$, $B$, or $C$? What if you change the position, length, or direction of the directed line segment $overline$?


    Videos
    R.1 Solving Equations

R.3 The Slope-Intercept Form of a Line

R.4 Solving a System of Equations

R.5 Multiplying Polynomials

R.10 Area and Perimeter Fundamentals

1.3 Rates, Ratios, and Proportions

1.4 Translation in a Coordinate Plane

1.6 Reflection, Rotation, and Symmetry

1.7 Composition of Transformations

2. Similar Figures and Dilation

2.2 Dilation and Similar Figures

2.3 Similarity, Polygons, and Circles

2.4 Similarity and Transformations

3.2 Conditional Statements

4. Parallel and Perpendicular Lines

4.2 More on Parallel Lines and Angles

4.4 Parallel Lines, Perpendicular Lines, and Slope

4.5 Parallel Lines and Triangles

    Videos
    5.1 Isosceles and Equilateral Triangles

5.3 Proving Triangles Congruent with SSS and SAS

5.4 Proving Triangles Congruent with ASA and AAS

6. Relationships Within Triangles

6.2 Perpendicular and Angle Bisectors

6.4 Centroids and Orthocenters

6.6 Optional: Inequalities in One Triangle

6.7 Optional: Indirect Reasoning

7. Similarity and Trigonometry

7.2 Similar Triangles: Side-Angle-Side Theorem

7.4 Similar Right Triangles

7.5 Special Right Triangles

7.7 Optional: Inverses of Trigonometric Functions

7.8 Law of Cosines and Law of Sines

8.4 More on Chords and Angles

    Videos
    9.1 Parallelograms and Their Diagonals

9.2 Deciding If a Parallelogram Is Also a Rectangle, Square, or Rhombus

9.3 Deciding If a Quadrilateral Is a Parallelogram

9.4 Optional: Polygons and Their Angles

9.6 Areas and the Coordinate Plane

9.7 Area of Regular Polygons

    Videos
    10.1 Three-Dimensional Figures, Cross-Sections, and Drawings


1.5: Rotations and Reflections of Angles - Mathematics

3844 days since
Winter Break

TEACHERS-STUDENTS RESOURCES

TRANSLATION & REFLECTION

GEOMETRY TRANSFORMATION

objective: Students will be able to

Students will be able to identify and compare the three congruence transformations,

apply the three congruence transformations to coordinates of the vertices of figures,

identify and apply dilations, and apply transformations to real-world situations.

- represent /draw and interpret the results of transformations and successive

transformations on figures in the coordinate plane.

• rotations (90°, 180°, clockwise and counterclockwise about the origin)

- identify locations, apply transformations, and describe relationships using coordinate geometry.

Compare transformations that preserve distance and angles to those that do not.

Solve problems involving transformations in order to solve real-world problem.

Ability to make connections between function transformations (F.BF.3) and geometric

 Knowledge that rigid transformations preserve the shape of a figure

Ability to use appropriate vocabulary to describe the rotations and reflections

 Ability to use the characteristics of a figure to determine and then describe what happens to the

figure as it is rotated (such as axis of symmetry, congruent angles or sides….

Ability to interpret and perform a given sequence of transformations and draw the result

 Ability to accurately use geometric vocabulary to describe the sequence of transformations that

will carry a given figure onto another

Using transformational geometry, create a reflection, translation, rotation, glide reflection and dilation of a

figure and apply transformations and use symmetry to analyze mathematical situations

 Experiment with transformations in the plane

Cluster Note: Build on student experience with rigid motions from earlier grades. Point out the basis of rigid

motions in geometric concepts, e.g., translations move points a specified distance along a line parallel to a specified line rotations move objects along a circular arc with a specified center through a specified angle

G.CO.2 Represent transformations in the plane using, e.g., transparencies and geometry software describe

transformations as functions that take points in the plane as inputs and give other points as outputs. Compare

transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch)

Make geometric constructions

G.CO.3 Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.

G.CO.4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments

G.CO.5 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.

What does "transform" mean, and what does it enable us to understand?

Why is it important to be able to

What are the similarities and differences between the images and pre-images

generated by translations?

What is the relationship between the coordinates of the vertices of a figure and the

coordinates of the vertices of the figure’s image generated by translations?

How can translations be applied to real-world situations?

Manipulation of geometric figures can

be a useful tool in real world situations

image preimage or original

translation translation vector

angle of rotation center of rotation

line of reflection rigid transformation

Worksheets, protractor, ruler, patty paper, Mira™

Optional – Dynamic geometry softwar

WARM UP: Use the matrices below to answer the question ,

Which matrix represents the expression ?

Reflection over a line k (notation rk ) is a transformation in which each point of the original figure ( pre-image ) has an image that is the same distance from the line of reflection as the original point but is on the opposite side of the line. Remember that a reflection is a flip . Under a reflection, the figure does not change size.
The line of reflection is the perpendicular bisector of the segment joining every point and its image.

A line reflection creates a figure that is congruent to the original figure and is called an isometry (a transformation that preserves length). Since naming (lettering) the figure in a reflection requires changing the order of the letters (such as from clockwise to counterclockwise), a reflection is more specifically called a non-direct or opposite isometry.

Properties preserved (invariant) under a line reflection:
1. distance (lengths of segments are the same) 2. angle measures (remain the same)
3. parallelism (parallel lines remain parallel)
4. colinearity (points stay on the same lines)
5. midpoint (midpoints remain the same in each figure)
6. orientation (lettering order NOT preserved. Order is reversed.)

Reflecting over the x -axis : (the x-axis as the line of reflection)

When you reflect a point across the x-axis, the x-coordinate remains the same,but the y-coordinate is transformed into its opposite.

The reflection of point across the x-axis is point .

Hint: If you forget the rules for reflections when graphing, simply fold your graph paper along the line of reflection (in this example the x-axis) to see where your new figure will be located. Or you can measure how far your points are away from the line of reflection to locate your new image. Such processes will allow you to see what is happening to the coordinates and help you remember the rule.

Reflecting over : (parallel to x-axis)

When you reflect a point across , the x-coordinate remains the same,but the y-coordinate is transformed into 2k-y.

The reflection of point across the x-axis is point .

Reflecting over the y -axis : (the y-axis as the line of reflection)

When you reflect a point across the y-axis, the y-coordinate remains the same, but the x-coordinate is transformed into its opposite. The reflection of point across the y-axis is point .

Reflecting over the : (parallel to the y-axis )

When you reflect a point across , the y-coordinate remains the same, but the x-coordinate is transformed into . The reflection of point across the y-axis is point .

Reflecting thru the origin also call 180 degree rotation

When you reflect a point across the origin , the y-coordinate is transformed into its opposite , and the x-coordinate is transformed into its opposite. basically, just change signs. The reflection of point across the origin is point .

Reflecting thru a different point The reflection of point across a point is point .

Reflecting over the line y = x or y = -x :

(the lines y = x or y = -x as the lines of reflection) When you reflect a point across the line y = x, the x-coordinate and the y-coordinate change places. When you reflect a point across the line y = -x, the x-coordinate and the y-coordinate change places and are negated (the signs are changed).

The reflection of the point across the line is the point .

The reflection of the point across the line is the point .

Reflecting over any line:

Each point of a reflected image is the same distance from the line of reflection as the corresponding point of the original figure. In other words, the line of reflection lies directly in the middle between the figure and its image -- it is the perpendicular bisector of the segment joining any point to its image. Keep this idea in mind when working with lines of reflections that are neither the x-axis nor the y-axis.

Each point of the original figure and its image are the same distance away from the line of reflection (which can be easily counted in this diagram since the line of reflection is vertical).

Flip order of x and y. Change signs according to what quadrant it's in.

Then the rotation of the point M about an angle about the origin maps it onto a point such that which is the rotation of the matrix about the origin through an angle .
To rotate about a point that is not the origin, first you move all the points so the center is the origin, use the usual rotation matrix, and then move all the points back to where you found them.

Example: If is rotated about an angle , determine the image point.

here , , . Rotation matrix is given by

Substituting the above values

Therefore, the image point is given by

INDEPENDENT PRACTICE

Graph the image of the figure using the transformation given

1) reflection across the x-axis

2) reflection across y = 3

reflection across y = 1

reflection across the x-axis

reflection across the x-axis

reflection across y = −2

Write a rule to describe each transformation

Draw the image according to the rule and identify the type of transformation

Complete the ordered pair rule that transforms each triangle to its image. Identify the transformation. Find all missing coordinates.

BCR: An air traffic control system at Little Rock National (LIT) airport, located at (2, 3) on the grid below, uses a radar system that sends out signals to determine the locations of airplanes. This system can detect planes within a circular region having a radius of 35 miles from LIT. Each grid unit represents 5 miles.

An airplane is heading directly toward LIT from the location represented by coordinates on the grid.

1. Can the plane be detected by the radar? Support your answer with mathematical evidence.

2. The air traffic controller instructs the pilot to begin circling the airport halfway between the airport

and her current location. What will be the coordinates of the plane’s location when the pilot begins

to circle the airport? Show your work or explain your answer.

1) is shown below

If is rotated clockwise about the origin, what will be the coordinates of the image of point T?

2) Use the graph below to answer question

Which graph shows a reflection across the x-axis of the image above?

3) The figure below is rotated clockwise about the origin, then reflected across the y-axis.

4) Josh is designing a cover for a paperback book. He is going to use the graphic shown above. He plans to reflect the graphic over the y-axis. What will be the coordinates of the reflection of point A?

5) Roberto is a computer graphics designer and is working on an ad for the local coffee shop. The figure above shows a coffee mug in two different positions. Which describes the transformation of the coffee mug in position I to the image in position II?

A. a reflection over a horizontal line and a translation down

* B. translation down and a reflection over a vertical line C. 180° rotation

D. translation to the right and a reflection over a vertical line

6) The figure graphed below is rotated clockwise about the origin and translated up 1 unit.

Which is the resulting image?

7) Which image will result from the figure below being rotated 90q clockwise about the origin and then

reflected across the y-axis?

8) Polygon STUVW is shown below.

After polygon STUVW is reflected across the y-axis, what are the coordinates of S′, the image of point S

9) The figure below is translated 3 units to the right, then 5 units down, and finally reflected over the x-axis.

What are the coordinates of the image of point X after the transformations?

If is rotated clockwise about the origin, what will be the coordinates of the

11) If the parallelogram below were translated 3 units left and 6 units down, what would be

the coordinates of the new image W′X′Y′Z′?

A. W′(–2, –1), X′(0, 3), Y′(5, 3), Z′(3, –1) B. W′(–1, –2), X′(3, 0), Y′(3, 5), Z′(–1, 3)

C. W′(4, –1), X′(6, 3), Y′(11, 3), Z′(9, –1) D. W′(7, 8), X′(9, 12), Y′(14, 12), Z′(12, 8)

12) Triangle JKL is translated 4 units left and 5 units up. What are the coordinates of the

A. (2, 6) B. (3, –3) * C. (– 6, 6) D. (–2, 6)

13) What would the figure below look like if it were reflected over the x-axis?

14) Triangle QRS is shown in the graph below.

Which of the following graphs shows ∆QRS rotated 90 degrees counterclockwise about the origin?

15) Segment JK is reflected across the y-axis to form . What are the coordinates of J′ and K′

A. J′(– 4, –5), K′(–2, 1) B. J′(5, – 4), K′(–1, 2)

16) The arrow above represents the needle on a compass. The needle is rotated 180° in the clockwise direction. What are the coordinates of point A after the rotation?

17) Triangle PQR has vertices of P(–2, –1), Q(1, 6), and R(3, –2). What are the coordinates of the vertices of the image of ∆PQR if the figure is translated 4 units right and 3 units up?

A. P′(2, 2), Q′(5, 9), R′(7, 1) B. P′(1, 3), Q′(4, 10), R′(0, 2)

C. P′(– 6, 2), Q′(–3, 9), R′(–1, 1) D. P′(– 8, –3), Q′(4, 18), R′(12, – 6)

18) The point of the heart (H) has a coordinate of (–5, –7) as shown above. The heart is reflected over the y-axis and then reflected over the x-axis. After both reflections, what are the coordinates of the point H?

19) The polygon above is the mapping of a school building. What translation rule moves point A

20) Which transformation describes the change from Figure M to Figure N?

A. dilation * B. reflection C. rotation D. translation

21) Sacha planned a fabric design by reflecting the triangle shown above over the x-axis. Which

list of coordinates represents the vertices of the triangle reflected over the x-axis?

A. (–2, –3), (–4, –6), (–8, 1) B. (–2, 3) , (–4, 6), (–8, 1)

C. (2, –3), (4, –6), (8, –1) D. (3, 2), (6, 4), (1, 8)

22) Which would move Flag A to Flag B in the graph below?

*A. clockwise rotation of 180° B. reflection over x-axis, then clockwise rotation of 90°

C. translation 8 units left and 7 units down

D. reflection over y-axis, then translation 8 units left and 7 units down

23) To plan a scene in an animated movie, Roger rotates the below figure around point P by 90° in a clockwise direction. Which drawing shows the pre-image and the final image?

24) A quilt design is formed by translating a polygon across the coordinate plane as shown in the figure below. Which is a translational rule that will translate point A to point B?

A. B. C. * D.

25) After this translation of point T, what are the coordinates of the new point?

A. (–1, 3) B. (0, 4) C. (3, –2) D. (5, –1)

26) Janelle and Franz are playing a game. The rule of the game is each playing piece must go through one transformation. Janelle reflects piece X over line 3. Where will piece X land?

A. on piece A * B. on piece B C. on piece C D. on piece D

27) Which graph represents the figure below reflected across the y-axis?

28) The coordinates of C are (--4, 1). Which translation moves to ?

A. translate 8 units left, 4 units up B. translate 3 units left, 8 units up

C. translate 3 units right, 8 units down * D. translate 8 units right, 4 units down

29) The point on the grid is reflected across the y-axis. What are its new coordinates?

A. (--5, 2) B. (--2, 5) C. (2, --5) D. (2, 5)

30) After translating the point (x, y) four units to the right, what are its new coordinates?

A. B. C. D.

31) Which figure represents a rotation of figure 1?

A. figure 2 * B. figure 3 C. figure 4 D. figure 5

32) What are the coordinates of PQRS after a translation of 6 units to the right and 6 units up?

33)Triangle ABC is reflected over the x-axis. What will be the coordinates of A', B', and C'?

34) Which shadow shows a reflection of the corresponding figure?

35)Which of the following figures represents a clockwise rotation of the flag about point A?


Rotations

High School Math based on the topics required for the Regents Exam conducted by NYSED.

Geometry Rotation
A rotation is an isometric transformation: the original figure and the image are congruent. The orientation of the image also stays the same, unlike reflections. To perform a geometry rotation, we first need to know the point of rotation, the angle of rotation, and a direction (either clockwise or counterclockwise). A rotation is also the same as a composition of reflections over intersecting lines.

The following figures show rotation about a point and rotation as a double reflection.

Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

We welcome your feedback, comments and questions about this site or page. Please submit your feedback or enquiries via our Feedback page.


What is a Translation?

In addition to reflecting or rotating an object, we can also translate the object to another place on the coordinate plane. Translation is the act of "sliding" our point or shape along the coordinate plane in a particular direction.

The shape can be translated up or down (or both!) any amount of distance along the plane. It maintains its shape and bearing, but is simply located elsewhere in the plane.

The way to notate that a translation is to occur is to say:

This means that your final coordinates for this point will be:

What is the new point for $T_<5,−2>(−3,6)$?

We know that we must add together our translated points to the corresponding $x$ and $y$ values of our original coordinates. So:

Our new coordinates for this point are at $(2, 4)$

You can see why this is true if we look at it on a graph.

Here, we have our starting point of $(-3, 6)$.

Now, we are moving positively (to the right) 5 spaces and negatively (downwards) 3 spaces. If we started at $(-3, 6)$, this wll put our new point at $(2, 4)$.

Our final answer is B , $(2, 4)$.


Rotational Symmetry

Lastly, a figure in a plane has rotational symmetry if the figure can be mapped onto itself by a rotation of 180° or less. This means that if we turn an object 180° or less, the new image will look the same as the original preimage. And when describing rotational symmetry, it is always helpful to identify the order of rotations and the magnitude of rotations.

The order of rotations is the number of times we can turn the object to create symmetry, and the magnitude of rotations is the angle in degree for each turn, as nicely stated by Math Bits Notebook.

In the video that follows, you’ll look at how to:

  1. Describe and graph rotational symmetry.
  2. Describe the rotational transformation that maps after two successive reflections over intersecting lines.
  3. Identify whether or not a shape can be mapped onto itself using rotational symmetry.

Translation
A translation moves a shape. A translation is a slide of a shape: without rotating, reflecting or resizing it.
Each point on the shape moves the same direction and the same distance.

Translate a Shape
To translate a shape, break the translation down into:

- how far we move the shape in a horizontal direction (left or right).
- how far we move the shape in a vertical direction (up or down).

Use a column vector to describe how far to move the shape in these directions.

- find how far a shape has moved in a horizontal direction (left or right).
- find how far a shape has moved in a vertical direction (up or down).

Write these in a column vector.


Rotation

In geometry, a rotation is a type of transformation where a shape or geometric figure is turned around a fixed point. It may also be referred to as a turn. A rotation is a type of rigid transformation, which means that the size and shape of the figure does not change the figures are congruent before and after the transformation. Below are two examples.

In the figure above, the wind rotates the blades of a windmill. On the right, a parallelogram rotates around the red dot.

The term "preimage" is used to describe a geometric figure before it has been transformed and the term "image" is used to describe it after it has been transformed. For 2D figures, a rotation turns each point on a preimage around a fixed point, called the center of rotation, a given angle measure. Two Triangles are rotated around point R in the figure below. For 3D figures, a rotation turns each point on a figure around a line or axis.


Rotational Symmetry

These lessons help Geometry students learn about rotational symmetry, with video lessons, examples and solutions.

In these lessons, we will learn

  • what is rotational symmetry?
  • how to find the order of rotation.
  • how to find the angle of rotation.

The following table gives the order of rotational symmetry for parallelogram, regular polygon, rhombus, circle, trapezium, kite. Scroll down the page for examples and solutions.

What Is Rotational Symmetry?

Symmetry in a figure exists if there is a reflection, rotation, or translation that can be performed and the image is identical. Rotational symmetry exists when the figure can be rotated and the image is identical to the original.

A regular polygon has a degree of rotational symmetry equal to its number of sides.

What Is The Order Of Rotation And Angle Of Rotation?

A figure has rotational symmetry if it coincides with itself in a rotation less than 360°.

The order of rotation of a figure is the number of times it coincides with itself in a rotation of 360°.

The angle of rotation for a regular figure is 360 divided by the order of rotation.

How To Find The Order Of Rotational Symmetry Of A Shape?

The order of rotational symmetry is the number of times you can rotate a shape so that it looks the same. The original position is counted only once (i.e. not when it returns to its original position)

The order of rotational symmetry of a regular polygon is the same as the number of sides of the polygon.

You can also deduce the order of rotational symmetry by knowing the smallest angle you can rotate the shape through to look the same.
180° = order 2,
120° = order 3,
90° = order 4.

The product of the angle and the order would be 360°.

How to relate between a reflection and a rotation and examine rotational symmetry within an individual figure

The following video will give the solutions for the Rotations, Reflections and Symmetry Worksheet. (Common Core, Geometry Lesson 15, Module 1)

Opening Exercise
The original triangle, labeled A, has been reflected across the first line, resulting in the image labeled B. Reflect the image across the second line. Carlos looked at the image of the reflection across the second line and said, “That’s not the image of triangle A after two reflections that’s the image of triangle A after a rotation!” Do you agree? Why or why not?

Discussion
When you reflect a figure across a line, the original figure and its image share a line of symmetry, which we have called the line of reflection. When you reflect a figure across a line and then reflect the image across a line that intersects the first line, your final image is a rotation of the original figure. The center of rotation is the point at which the two lines of reflection intersect. The angle of rotation is determined by connecting the center of rotation to a pair of corresponding vertices on the original figure and the final image. The figure above is a 210° rotation (or 150° clockwise rotation).

Exploratory Challenge
Line of symmetry of a figure: This is an isosceles triangle. By definition, an isosceles triangle has at least two congruent sides. A line of symmetry of the triangle can be drawn from the top vertex to the midpoint of the base, decomposing the original triangle into two congruent right triangles. This line of symmetry can be thought of as a reflection across itself that takes the isosceles triangle to itself. Every point of the triangle on one side of the line of symmetry has a corresponding point on the triangle on the other side of the line of symmetry, given by reflecting the point across the line. In particular, the line of symmetry is equidistant from all corresponding pairs of points. Another way of thinking about line symmetry is that a figure has line symmetry if there exists a line (or lines) such that the image of the figure when reflected over the line is itself.

Does every figure have a line of symmetry?

How to find the angle of rotation for regular polygons?

The angle of rotation of a regular polygon is equal to 360° divided by the number of sides.

Rotational Symmetry

The order of Rotational Symmetry tells us how many times a shape looks the same when it rotate 360 degrees. Determine the order of rotational symmetry for a square, a rectangle and an equilateral triangle.

Basic Rotational Symmetry

Introduction to rotational symmetry with fun shapes.

Rotational Symmetry

Learn to identify and describe rotational symmetry.

Tell whether each figure has rotational symmetry. If it does, find the smallest fraction of a full turn needed for it to look the same.

How many times will the figure show rotational symmetry within one full rotation?

Also, identify the degree of rotational symmetry.

Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

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