10r geometry transformations warm-up page warm-up day 9.1: find the distance …notes... · ·...

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10R Geometry Transformations Warm-up Page Warm-Up Day 9.1: Find the distance between points A and B if A (-3, 10) and B(4, 2). Express your answer in simplest radical form. Warm-up Day 9.2: The coordinate of the vertices of quadrilateral JAKE are J(0, 3a), A(3a, 3a), K(4a, 0), and E(-a, 0). Prove that JAKE is an isosceles trapezoid. Warm-up Day 9.3: For each of the following, state the line/point of reflection

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Warm-up Day 9.4 Warm-up Day 9.5

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10R Geometry Transformations Classwork Day 9.1 Translations

Vocabulary: Isometry: A transformation that preserves ________________________ and _________________________ AKA: Congruence Transformation: Location changes but _______________________________________ Translations: Moves every point of a figure the same distance in the same direction. TRANSLATIONS ARE _______________________________FROM____________________________ Notation: Option 1:___________________________________ where

_____ shows how to move __________________________ and

_____ shows how to move __________________________

Option 2:_____________________________________________ Preimage:__________________________ Image:_____________________________ **Label Everything!!! Example: Graph triangle ABC with vertices of A(-3, 4), B(1, 0), and C(2, 3). Find the image of ABC after the translation (x,y)⇒(x+2, y -4).

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Example 2: Given quadrilateral ABCD, where A(2, 2), B(2, 5), C(4, 5), and D(4, 2), translate ABCD 5 units down and 4 units left. 1. Express the translation using T(n,m) notation. 2. A(2, 2) ⇒ A'(_____, _____) B(2, 5) ⇒ B'(_____, _____) C(4, 5) ⇒ C'(_____, _____) D(4, 2) ⇒ D'(_____, _____) 3. Show that AB ≅ A'B' Vector Vocab: Vector: Has both __________________________. Represented with an arrow drawn from preimage to image. Initial Point: _____________________________ Terminal Point: ___________________________ Component Form: combination of horizontal and vertical components (movement). Notation:_____________ Ex. What would the component form be for ray AB?

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Practice: Use the point P(-4, 7). Find the component form of the vector that describes the translation to P'. 1. P'(-7, 2) 2. P' (-6, 10) 3. What is the image of (x,y) after a translation of 3 units right and 7 units down? (1) (x + 3,y – 7) (3) (x – 3,y – 7) (2) (x + 3,y + 7) (4) (x – 3,y + 7) 4. A polygon is transformed according to the rule: (x, y) -> (x+2, y) Every point of the polygon moves two

units in which direction? (1) up (3) left (2) down (4) right 5. A design was constructed by using two rectangles ABDC and A’B’D’C’. Rectangle A’B’D’C’ is the result

of a translation of rectangle ABDC. The table of translations is shown below. Find the coordinates of points B and D’.

Review of Translation Notation: ΔA'B'C' is the image of ΔABC after a translation. Write the rule for the translation. 1. T-Form: _________________________________ 2. (x,y) Form: ______________________________ 3. Component Form: _________________________ 4. Sentence Form: ___________________________

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Name: _______________________________________ Date: ________________________ HW 9.1: Translations Math 10R Period ________

1. _____ If x = –2 and y = –1, which point on the accompanying set of axes

represents the translation (x,y) (x + 2, y - 3)→ ? (1) Q (3) S (2) R (4) T 2. _____ What is the image of point (2,5) under the translation that shifts (x,y)

to ?)2,3( −+ yx (1) (0,3) (3) (5,3) (2) (0,8) (4) (5,8) 3. _____ What are the coordinates of P′, the image of P (–4, 0) under the translation ?)6,3( +− yx (1) (–7,6) (3) (1,6) (2) (7,–6) (4) (2,–3) 4. _____ A translation moves P(3,5) to 'P (6,1). What are the coordinates of the image of point (–3,–5) under

the same translation? (1) (0,–9) (3) (–6,–1) (2) (–5,–3) (4) (–6,–9) 5. _____ The image of the origin under a certain translation is the point (2,-6). The image of point (-3,-2) under

the same translation is the point

(1) (-6,12) (3) )31,

23(−

(2) (-5,4) (4) (-1,-8) 6. Triangle A’B’C’ is the image of triangle ABC 7. Solve for each variable. after a translation. Write a rule for the translation.

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8. Two parabolic arches are to be built. The equation of the first arch can be expressed as y x= − +2 9, with a range of 0 ≤ y ≤ 9, and the second arch is created by the transformation T7,0. On the accompanying set of axes, graph the equations of the two arches. Graph the line of symmetry formed by the parabola and its transformation and label it with the proper equation.

9. The vertices of a rectangle are Q(2, -3), R(2, 4), S(5, 4), and T(5, -3) a. Plot rectangle QRST. b. Translate QRST 3 units left and 2 units up. c. Find the area of Q’R’S’T’

d. Will the area of Q’R’S’T’ be the same as QRST?

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10R Geometry Transformations Classwork 9.2: Reflections

Vocabulary Reflections:______________________________________________________________________________ Notation:_____________________ Reflections are:________________________________ Line Reflections 1) If (a, b) is reflected over the x-axis, its image is ___________ Rx-axis(x, y)=________________________ 2) If (a, b) is reflected over the y-axis, its image is ___________ x & y axis Ry-axis(x, y)=________________________ 3) If (a, b) is reflected over y=x, its image is ___________ Ry=x(x, y)=________________________ y = x 4) If (a, b) is reflected over y = -x, its image is ___________ Ry=-x(x, y)=________________________ Y = -x

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Line Reflections over any line Example: Reflect ∆ABC over the line y = 2 Important Information about Line Reflections! 1. The line (where a mirror may be placed) is called the _________________________ 2. The distance from a point to the line of reflection is the same as the distance from the point's

______________________________________________________________________________.

3. A reflection can be thought of as _______________________________ an object over the line of reflection. 4. The line of reflection is the perpendicular bisector of the segment joining every point __________________

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Point Reflections:__________________________________________________________________________ For every point in the figure, there is another point found

_____________________ it on the other side of the center such

that the point of reflection becomes the _________________

of the segment joining the point with its image.

Under a point reflection, figures do not change size.

Isometry?____________________________________

POINT REFLECTIONS

Notation:

If (a, b) is reflected through the origin, its image is __________

rorigin (x, y) = _________

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Name: _______________________________________ Date: ________________________ HW 9.2: Reflections Math 10R Period ________

1. _____ What is the image of point (–3,–1) under a reflection in the origin?

(1) (3,1) (3) (1,3) (2) (–3,1) (4) (–1,–3)

2. _____ Ms. Brewer’s art class is drawing reflected images. She wants her students to draw images

reflected in a line. Which diagram represents a correctly drawn image?

3. _____ Point (-2, 3) is reflected in the x-axis. In which quadrant does its image lie?

(1) I (3) III (2) II (4) IV

4. _____ The parabola shown in the accompanying diagram undergoes a reflection in the y-axis. What will be the coordinates of the turning point after the reflection?

(1) (3,-1) (3) (-3,1) (2) (3,1) (4) (-3,-1)

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5. _____ A function, f, is defined by the set {(2,3), (4,7), (-1,5)}. If f is reflected in the line y = x, which

point will be in the reflection? (1) (5,-1) (3) (1,-5) (2) (-5,1) (4) (-1,5)

6. In the accompanying diagram of square ABCD, F is the midpoint of AB, G is the midpoint of BC, H is the

midpoint of CD , and E is the midpoint of DA. a) Find the image of ΔEOA after it is reflected in line . _________

b) Is this isometry direct or opposite? ____________________

Explain your answer.

7. On the accompanying grid, draw and label quadrilateral ABCD with points A(1,2), B(6,1), C(7,6), and D(2,7).

a. On the same set of axes, plot and label quadrilateral A'B'C'D' , the reflection of quadrilateral ABCD in the y-axis.

b. Determine the type of quadrilateral ABCD is by using coordinate geometry. c. Determine the area, in square units, of quadrilateral A'B'C'D' .

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10R Geometry Transformations Classwork 9.3: Rotations

Vocabulary: Rotations:_________________________________________________________________________________ Notation:_____________________________________________ Rotations are __________________________________________ Clockwise: Counter Clockwise: ***Rotations are always ___________________________, unless told otherwise (by neg) Isometry?_____________________________________ ROTATIONS Notation: 1) If (a, b) is rotated about the origin 90o, its image is __________

R90 (x, y) = _________

2) If (a, b) is rotated about the origin 180o, its image is __________

R180 (x, y) = _________

3) If (a, b) is rotated about the origin 270o, its image is __________ R270 (x, y) = _________

How to do a rotation other than 90, 180, or 270 1. Draw a segment from the point about which the figure is being rotated, to the figure. 2. Using a protractor, measure the angle required. 3. Make sure both segments drawn are congruent. 4. Connect each vertex. Example: R120 (x, y)

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Example: Graph quadrilateral RSTU with vertices R(3, 1), S(5, 1), T(5, -3), and U(2, -1). Then perform the following transformation: R270 Give the point of each image for the given rotation & state the quadrant of the image. 1. R90 (-3, 10) = __________ ___________ 2. R270 (-2, -1) = __________ ___________ 3. R180 (5, 15) = __________ ___________ 4. R270 (-4, 8) = __________ ___________ 5. R90 (4, -2) = __________ ___________

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Name: _______________________________________ Date: ________________________ HW 9.3: Rotations Math 10R Period ________

1. _____ If point (5,2) is rotated counterclockwise 90° about the origin, its image will be point (1) (2, 5) (3) (-2, 5) (2) (2, -5) (4) (-5, -2) 2. _____ The accompanying diagram shows the starting position of the spinner on a board game.

How does this spinner appear after a 270° counterclockwise rotation about point P? (2) (4) 3. _____ Point (-2, 3) is rotated 180 degrees about the origin. In which quadrant does its image lie? (1) I (3) III (2) II (4) IV 4. _____ Rectangle PQRS is rotated 90 degrees clockwise about point O. Find the image of PS. (1) AD (3) QR (2) QP (4) AB

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5. _____ Identify the coordinates of point (-6, 5) under a rotation of 180 degrees about the origin. (1) (6, 5) (3) (-6,-5) (2) (5,-6) (4) (6,-5) 6. _____ Which transformation is an example of an opposite isometry? (1) (x,y) → (x + 3,y – 6) (3) (x,y) → (y,x) (2) (x,y) → (3x,3y) (4) (x,y) → (y,–x)

7. On the accompanying grid, draw and label quadrilateral ABCD with points A(1,2), B(6,1), C(7,6), and D(3,7).

a. On the same set of axes, plot and label quadrilateral A'B'C'D' , the rotation of quadrilateral ABCD 270º about the origin.

8. Triangle ABC is shown on the accompanying grid.

a) State the points of its image after ABC is rotated 90º counter clockwise and then 180º clockwise. b) Find the slope of AB and A’’B’’.

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10R Geometry Transformations Classwork 9.4: Dilations

Vocabulary Dilations:_________________________________________________________________________________ Notation:______________________________________________________ Dilations are_______________________________________ by ________________________________ Isometry?_________________________________- Example: 1) If (a, b) is dilated by a scale of 3, its image is __________

D3 (x, y) = _________

2) If (a, b) is dilated by a scale of ½, its image is __________

D1/2 (x, y) = _________

Examples: State the coordinates of the image given the pre-image ordered pair. 1. D2(3, 5) = __________________ 3. D0.2(-8, -10) = ________________ 2. D-2(-4, 12) = ________________ 4. D3.5(-6, 3) = ________________

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Name: _______________________________________ Date: ___________________________ HW: 9.4- Dilations w/ Compositions Math 10R Period __________

1. T –2, 0 (-3, 5) ( ___ , ___ )

2. T a, b (-3, 5) ( -3, 1 ) a, b = _____________ 3. Which transformation represents a dilation?

(1) (8,4) (11,7)→ (3) )8,4()4,8( −−→ (2) (8,4) ( 8, 4)→ − (4) )2,4()4,8( →

4. T –3, 1 ( _____, _____ ) (4, -2)

5. Use the rule (x, y) (x –3, y + 12) to find the image of (2, 0)

6. Find the rule for the translation so the image of A is A’: A(-1, 2) A’(-2, -3)

7. Under a dilation with respect to the origin, the image of P( , )−15 6 is P' ( , ).−5 2 What is the constant of dilation?

(1) -4 (3) 3

(2) 13

(4) 10

8. What are the coordinates of P’, the image of P(-2, 4) after a counterclockwise rotation of 180° about the

origin.

9. Write the coordinates of the image of the point (-1, 3) under a rotation about the origin of 90° clockwise.

10. What are the coordinates of the image of B(-3, -4) after a rotation of 90° counterclockwise.

11. Find (x, y): ry=-3(x, y) (2, 1) (x, y) = __________

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12. What are the coordinates of the point that is the image of N(5, -3) under a reflection in the y-axis?

13. Find the image of (-4, 12) after being reflected over the y = x.

14. The vertices of ΔABC are A(-4, 7), B(3, -2), and C(8, -2). After a translation that maps (x, y) onto (x + h, y + k), ΔA’B’C’ is the image of ΔABC. a. If ΔA’B’C lies completely in Quadrant I, what are

the smallest possible integer values of h and k?

b. How many square units are in the area of ΔA’B’C?

15. Triangles ABC has coordinates A(1, 2), B(0,5), C(5, 4). a. Graph and label ΔABC.

b. Graph and label ΔA’B’C’ after a rotation of 270o.

c. Dilate ΔA’B’C’ by a scale factor of 2.

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10R Geometry Transformations Classwork 9.5: Glide Reflections

Glide Reflections: ________________(a slide or glide) and a ___________are performed one after the other **NOTE:_______________________________________________________________________________

Isometry?______________________

Example:

Complete the following: T-3, 0 o rx-axis

Complete the following: rx-axis o T-3, 0

Does the order of the composition make a difference?___________________________________________

**This transformation is ____________________________________________

**When reflecting in x-axis, y stays the same in the translation and when reflecting in y-axis, the x stays

the same in the translation.

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Given triangle A(-9,-3), B(-3,3), C(0,0), draw the image formed by a dilation with center at the origin and scale factor or 2/3, followed by a reflection in the y-axis, and followed by translation T1,1. 1. Express the above transformation using proper notation. 2. Compete the transformation of the graph provided.

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Name: ___________________________________________ Date: ________________________ HW: 9.5- Review #1 Math 10R Period ________

Part I: 1. ______ If the coordinates of point A are (–2,3), what is the image of A under r Dy axis− 3 ?

(1) (–6,–9) (3) (5,6) (2) (9,–6) (4) (6,9)

2. ______ What is the image of point (1,1) under r Rx axis− °0 90, ?

(1) (1,1) (3) (–1,1) (2) (1,–1) (4) (–1,–1)

3. ______ What are the coordinates of point A', the image of point A(-4,l) after the composite

transformation xyrR =°90 where the origin is the center of rotation?

(1) (-1,-4) (3) (1,4) (2) (-4,-1) (4) (4,1)

4. ______ The coordinates of ΔJRB are J ( , ),1 2− R( , ),−3 6 and B( , ).4 5 What are the coordinates of the

vertices of its image after the transformation T ry2 1, ?− −axis

(1) (3,1), (-1,-7), (6,-6) (3) (1,-3), (5,5), (-2,4) (2) (3,-3), (-1,5), (6,4) (4) (-1,-2), (3,6), (-4,5)

5.______ Which transformation is an example of an opposite isometry? (1) (x,y) → (x + 3,y – 6) (3) (x,y) → (y,x) (2) (x,y) → (3x,3y) (4) (x,y) → (y,–x)

6. ______ Which transformation of y x= 2 results in the function y x= −2 2 ?

(1) T0 1,− (3) ry−axis

(2) T0 2,− (4) rx−axis

7. ______ In the accompanying diagram, ΔA B C' ' ' is the image of ΔABC and Δ ΔA B C ABC' ' ' .≅

Which type of transformation is shown in the diagram? (1) line reflection (3) translation (2) rotation (4) dilation

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8. ______ Which transformation does not always result in an image that is congruent to the original figure? (1) dilation (3) rotation (2) reflection (4) translation

9. ______ Which transformation of the graph of 2xy = would result in the graph of ?22 += xy

(1) 2D (3) 2=yr

(2) 2,0T (4) 90,0R

10. ______ The perimeter of ΔA B C' ' ' , the image of ΔABC , is twice as large as the perimeter of ΔABC . What type of transformation has taken place? (1) dilation (3) rotation (2) translation (4) reflection

11. ______ The accompanying diagram shows a transformation.

Which transformation performed on figure 1 resulted in figure 2? (1) rotation (3) dilation (2) reflection (4) translation

12. ______ The transformation of ∆ABC to ∆AB’C’�is shown in the accompanying diagram. This transformation is an example of a

(1) line reflection in line (2) rotation about point A (3) dilation (4) translation

13. ______ Which transformation is not an isometry?

(1) ry x= (3) T3 6,

(2) R0 90, ° (4) D2

14. ______ Which transformation represents a dilation?

(1) (8,4) (11,7)→ (3) )8,4()4,8( −−→ (2) (8,4) ( 8, 4)→ − (4) )2,4()4,8( →

15. ______ Which image represents a line reflection?

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Part II/III/IV: 16. On the accompanying grid, graph and label AB , where A is (0,5) and B is (2,0). Under the transformation

),(ABrr axisyaxisx −− A maps to ′′A , and B maps to ′′B . Graph and label ′′ ′′A B . What single

transformation would map AB to ′′ ′′A B ? 17. The coordinates of the endpoints of AB are A(2,6) and B(4,2). Is the image '' ''A B the same if it is reflected

in the x-axis, then dilated by 12

as the image is if it is dilated by 12

, then reflected in the x-axis? Justify your

answer. #16 #17

18. On the accompanying set of axes, draw the reflection of ABCD in the y-axis.

a. Label and state the coordinates of the reflected figure.

b. Show that ABCD is congruent to A’B’C’D’

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19. 20. 21. Quadrilateral ABCD contains the points

A(2, -1), B(-3, 1), C(2, 4), and D(4, 4). a. Plot ABCD on the grid provided. b. Plot A’B’C’D’ after a dilation of 2. c. Show that ABCD is similar to

A’B’C’D’

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10r geometry transformations warm-up page warm-up day 9.1: find the distance …notes... · ·...

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