Upload
others
View
7
Download
0
Embed Size (px)
Dilations: Practice Name_____________________________________ Geometry section 9-5 Date_______________________Period__________ Find the image of each point under each of the following transformations. (Do all four transformations for each point.) Point (x, y) → ( 1
4x, 1
4y) (x, y) → (x, 3y) (x, y) → (4x, 4y) (x, y) → (0.5x, 0.5y)
1. (8, 4)
2. (-12, 3)
3. (-2, 2)
4. (0, 0)
State the mapping notation for the stretching, shrinking, expansion or contraction that maps the given point into the image point shown. Also, describe the transformation, i.e. horizontal shrink, vertical stretch, etc. Mapping Mapping Notation Description
5. (5,3) → (5,6)
6. (3,2) → (6,4)
7. (10,-2) → (5,-2)
8. (8,-8) → (4,-4)
9. (-5,8) → ( 25− ,4)
10. (-6,-9) → (-4,-6)
Find the images of A(0,1), B(-3,2), C(-4,0), and D(2,-3). Draw ABCD. In another color, draw A′B′C′D′.
11. (x, y) → (2x, 2y) 12. (x, y) → (0.5x, y) 13. (x, y) → (x, 3y)
14. Graph the points A(-1,2), B(2,2), C(1,-3), and D(-2,-3). Quadrilateral ABCD is mapped to quadrilateral A′B′C′D′ under a dilation. Graph A′(-2,4), B′(4,4), C′(2,-6), and D′(-4,-6).
What kind of dilation does this represent?
What is the scale factor from ABCD to A′B′C′D′?
Describe this dilation in functional notation. 15. Given the points A(-2, 4) and B(6, 10), find:
a) slope of b) equation of ABAB c) length of AB Using the functions in the first row below and the points A and B above, fill in the table.
(x, y) → (2x, 2y) (x, y) → ( 12
x, 12
y) (x, y) → (2x, y) (x, y) → (x, 12
y)
Using the transformation at the top of the column find A′ & B′
A′=(_____, _____)
B′=(_____._____)
A′=(_____, _____)
B′=(_____._____)
A′=(_____, _____)
B′=(_____._____)
A′=(_____, _____)
B′=(_____._____)
Find slope of A B′ ′
How does slope of A B′ ′ com re with slope of ?
paAB
W e the equation of
ritA B′ ′
How does thequation of compare to tequation of ?
e A B′ ′he AB
Find A′B′
How does A′B′ compare to AB?
Transformations Review Name: ___________________________ Date: ________________ Period: _____ 1. Given A(2, -6) and B(0, 0), find
(a) length of AB __________ (b) slope of AB __________ A. translation E. 180-degree rotation I. horizontal stretching B. reflection across the x-axis F. 270-degree rotation J. horizontal shrinking C. reflection across the y-axis G. vertical stretching K. expansion D. 90-degree rotation H. vertical shrinking L. contraction For each of the transformations described in column 1, answer the questions in row 1. Show work on another sheet of paper.
Letter from list above that corresponds.
Image of A(2, -6) and B(0, 0)
Describe any fixed points.
Length of ' 'A B . Is
distance preserved?
Slope of ' 'A B . Are AB and
' 'A B || , , or neither?
⊥
Would an image triangle under this transformation be the same size and/or shape as a preimage triangle?
2. (x, y) → (x +3, y)
3. x’ = 3x, y’ = y
4. (x, y) → (-x, -y)
5. (x, y) → (x, -y)
6. x’ = 2x, y’ = 2y
7. x’ = x – 5, y’ = y + 2
8. x’ = -y, y’ = x
9. (x, y) → (y, -x)
10. x’ = -x, y’ = y
11. x’ = 0.5x, y’ = 0.5y
12. (x, y) → (x, 4y)
Letter from list below that corresponds.
Image of A(2, -6) and B(0, 0)
Describe any fixed points.
Length of ' 'A B . Is
distance preserved?
Slope of ' 'A B . Are AB and
' 'A B || , , or neither?
⊥
Would an image triangle under this transformation be the same size and/or shape as a preimage triangle?
13. x’ = ¼ x, y’ = y
14. (x, y) → (x, 13
y)
15. x’ = 2x + 2 y’ = 3y – 1
A. translation E. 180-degree rotation I. horizontal stretching B. reflection across the x-axis F. 270-degree rotation J. horizontal shrinking C. reflection across the y-axis G. vertical stretching K. expansion D. 90-degree rotation H. vertical shrinking L. contraction 16. Which of the transformations A-L and #2-15 have NO fixed points? 17. Triangle KMP lies entirely in quadrant I. In which quadrant will its image lie after each of these:
b) reflection across the x-axis c) reflection across the y-axis d) 90-degree rotation e) 180-degree rotation f) 270-degree rotation k) expansion l) contraction m) reflection across the line y = x
18. Vocabulary:
A transformation in which the pre-image and image are congruent is a(n) ____________________.
A transformation in which the size is different but the shape is preserved is a(n) ______________.
A composition of two reflections across two parallel lines is a(n) ____________________.
A composition of two reflections across two intersecting lines is a(n) ____________________.
19. Using the graph to the right,
________________ Find the translation rule that maps T onto T’(4,6).
__________ Find the image of S after reflecting across the y-axis.
__________ Find the image of W after reflecting across the x-axis.
__________ Find the image of V after a 90° counterclockwise rotation.
__________ Find the image of U after dilating by a scale factor of 31 .
20. A O P X Z C D W
Which letters do not have reflectional symmetry? __________
Which letters have only 1 line of symmetry? __________
Which letters have 2 lines of symmetry? __________
Which letters do not have rotational symmetry? __________
Which letters have a 90° angle of rotation? __________
Which letters have a 180° angle of rotation? __________
Similar Figures: Proportions, Algebra, Name_______________________________ and Word Problems Date_______________________Pd_______ Find the missing variable and side lengths (if requested) for each pair of similar figures using proportions. Assume that all pairs of figures given are similar. Show all work. 2. c = ______
5. b = ______
7. q = ______
8. k = ______, m AB = ______, m AD = ______
9. j = ______
10. x = ______, m JK = ______
11. Jeremy’s house is 45 feet wide. In a photograph the width of the house was 2.5 inches, and its height was 2 inches. What is the actual height of Jeremy’s house?
12. Brandon wants to reduce a figure that is 9 inches tall and 16 inches wide so that it will fit on a 9-inch-by-12-inch piece of paper. If he reduces the figure proportionally, what is the maximum size the reduced figure could measure?
13. Troy used chalk to outline a triangular plot of land in his backyard. The plot of land has a perimeter of 26 feet, with its longest side measuring 8 feet 10 inches. Troy wants to outline a second triangular plot of land similar to the first but with a perimeter of 42 feet. What is the measure of the longest side of the second triangular plot of land?
14. Kate has 2 similar trapezoidal pieces of paper, as shown below. Using the dimensions given, find the perimeter of trapezoid LNQS.
10q + 15
4
60
3
A
C D 8 - k
4
2
j - 6 2j - 11
4
10 10
x
5 x - 3
E F
G H
J K
L M
A ~ABCΔ DEFΔ b + 5 24 3 4 B C
D c + 5
1 2b - 5
3
4k - 2
F E
Similar Figures: Practice on the Coordinate Plane Name______________________________________________ Date ________________________________ Period ________ Two of the three figures are similar. Name the two similar figures and give the scale factor. Dashed Dotted Solid
Example 1
solid to dashed with scale factor 4 (or ratio 1:4) or dashed to solid with scale factor 4
1 (or ratio 4:1)
1. 2.
3.
4.
5.
6.
7.
8.
9. In problem 1 the dimensions of the dotted triangle are _____ times those of the solid triangle. The area of the dotted triangle is ______ times that of the solid triangle. 10. In problem 2 the dimensions of the dotted triangle are _____times those of the dashed triangle. The area of the dotted triangle is _____ times that of the dashed triangle.
Name:__________________________ Period:___
Similar Triangles Investigations There are 3 ways to determine if triangles are similar: by their angles (Angle-Angle), their sides (Side-Side-Side) or a combination of sides and an included angle (Side-Angle-Side). For this activity we will need graph paper, pencil, TAKS formula chart and a protractor.
Investigation 1 (AA) Triangles with Two Pairs of Congruent Angles A B X Y
1. Using a protractor and a straight edge, draw a triangle ABC with a 50˚ angle at A∠ and a 60˚ angle at B∠ . 2. Using a protractor and a straight edge, draw triangle XYZ of a different size with a 50˚ angle at X∠ and a
60˚angle at Y∠ . Question: What do you notice about the shapes of the two triangles? Question: How can you determine the measure of the third angle without a protractor? What is its measure?
3. Using the centimeter ruler on your formula chart, measure the sides of each of the two triangles to the
nearest tenth of a centimeter and record the measures below. 4. Calculate the ratio of the length of each side of the triangle ABC to the length of the corresponding side of
triangle XYZ. Do this for all three sides of the two triangles. (Divide XYZ/ABC).
Triangle ABC
Length Triangle XYZ
Length Ratio
AB XY
BC YZ
AC XZ
Question: How do the three ratios compare? Question: Are the triangles similar? Justify your answer based on lengths and angles. Conjecture: If two angles of one triangle are congruent to two angles of another triangle, then the triangles are _____________ by ________________.
Name:__________________________ Period:___
Similar Triangles Investigations
There are 3 ways to determine if triangles are similar: by their angles (Angle-Angle), their sides (Side-Side-Side) or a combination of sides and an angle (Side-Angle-Side). For this activity we will need graph paper, pencil, TAKS formula chart and a protractor.
Investigation 2A (SSS) Triangles with Three Pairs of proportional sides
1. On graph paper, draw triangle ABC with A at (-2,-4), B at (0, 5) and C at (4, 3). 2. Dilate triangle ABC by a scale factor of 2 and label the points XYZ.
3. Using the centimeter ruler on your formula chart, measure the sides of each of the two triangles to the
nearest tenth of a centimeter and record the measures below. 4. Calculate the ratio of the length of each side of the triangle ABC to the length of the corresponding side of
triangle XYZ. Do this for all three sides of the two triangles. (Divide XYZ/ABC).
Triangle ABC
Length Triangle XYZ
Length Ratio
AB XY
BC YZ
AC XZ
Question: How do the three ratios compare? 5. Using a protractor, measure each angle and record the measures below.
Question: Are the triangles similar? Justify your answer based on lengths and angles. Conjecture: If three sides of one triangle are proportional to three sides of another triangle, then the triangles are _____________ by ________________.
Triangle ABC
Measure Triangle XYZ
Measure
A∠ X∠
B∠ Y∠
C∠ Z∠
Name:__________________________ Period:___
Similar Triangles Investigations
There are 3 ways to determine if triangles are similar: by their angles (Angle-Angle), their sides (Side-Side-Side) or a combination of sides and an angle (Side-Angle-Side). For this activity we will need graph paper, pencil, TAKS formula chart and a protractor.
Investigation 2B (SSS) Triangles with Three Pairs of proportional sides
1. On graph paper, draw triangle ABC with A at (-4,-8), B at (0, 10) and C at (8, 6). 2. Dilate triangle ABC by a scale factor of 1/2 and label the points XYZ.
3. Using the centimeter ruler on your formula chart, measure the sides of each of the two triangles to the
nearest tenth of a centimeter and record the measures below. 4. Calculate the ratio of the length of each side of the triangle ABC to the length of the corresponding side of
triangle XYZ. Do this for all three sides of the two triangles. (Divide XYZ/ABC).
Triangle ABC
Length Triangle XYZ
Length Ratio
AB XY
BC YZ
AC XZ
Question: How do the three ratios compare? 5. Using a protractor, measure each angle and record the measures below.
Triangle ABC
Measure Triangle XYZ
Measure
A∠ X∠
B∠ Y∠
C∠ Z∠
Question: Are the triangles similar? Justify your answer based on lengths and angles. Conjecture: If three sides of one triangle are proportional to three sides of another triangle, then the triangles are _____________ by ________________.
Name:__________________________ Period:___
Similar Triangles Investigations There are 3 ways to determine if triangles are similar: by their angles (Angle-Angle), their sides (Side-Side-Side) or a combination of sides and an angle (Side-Angle-Side). For this activity we will need graph paper, pencil, TAKS formula chart and a protractor.
Investigation 3 (SAS) Two Sides of a Triangle are Proportional to Two Corresponding Sides of Another Triangle and the Included Angle Congruent
1. On graph paper, draw triangle ABC with A at (-6,-7), B at (9, -3) and C at (9, 2). 2. Draw a vertical line that intersects side AB and side AC. Label the intersection of that line and side AB point Y
and label the intersection of that line and side AC point Z, creating triangle AYZ. Side YZ is parallel to side BC.
3. Using the centimeter ruler on your formula chart, measure the AB, AC, AY and AZ to the nearest tenth of a centimeter and record the measures below.
4. Calculate the ratio of the length of the sides of the triangle ABC to the length of the corresponding side of triangle AYZ. Do this for the following sides of the two triangles. (Divide AYZ/ABC).
Triangle ABC
Length Triangle AYZ
Length Ratio
AB AY
AC AZ
Question: Compare the ratios? Question: Did the measure of A∠ change? Question: Describe the position of A∠ in relationship to the proportional sides. 5. Using a protractor, measure each angle and record the measures below.
Triangle ABC
Measure Triangle AYZ
Measure
A∠ A∠
B∠ Y∠
C∠ Z∠
Question: Are the triangles similar? Justify your answer based on lengths and angles. Conjecture: If two sides of a triangle are proportional to two sides of another triangle and their included angles are congruent, then the triangles are _____________ by ________________.
Name:__________________________ Period:___
THE NEXT PAGE GOES ON THE BACK OF EVERY INVESTIGATION
Name:__________________________ Period:___
First let’s remember what similar means. For 2 polygons to be similar, what two conditions must be met: (1) all corresponding angles are ________________, and (2) all corresponding side lengths are ________________. However, there are 3 shortcuts for triangles! Now that you have worked with your group on your triangle investigation let’s summarize the investigations and answer questions on similar triangles. Note: the symbol for similarity is ~
Investigation 1 – AA Similarity
Conjecture: If two angles of one triangle are congruent to two angles of another triangle, then the triangles are _____________ by ________________.
1. ΔABC ∼ Δ______ by _______ (The order matters!!!) Investigation 2 – SSS Similarity Conjecture: If three sides of one triangle are proportional to three sides of another triangle, then the triangles are _____________ by ________________. 2. ΔHJK ∼ Δ______ by _______
Investigation 3 – SAS Similarity
Conjecture: If two sides of a triangle are proportional to two sides of another triangle and their included angles are congruent, then the triangles are _____________ by ________________. 3. ΔLMN ∼ Δ______ by _______ Note: Will SSA determine similarity?
Describe the difference between SAS and SSA?
A
B
C
40°
35°
40°
35° G F
L
J
H K
15 20
25
Q
P R
12 16
20
L M
N
8
12
U
S T
9
6
Justify:
Justify:
Justify:
Pg 385 # 2-18 evens, 11, 17, 21, 24-36 evens, 37, 44.
2. No, not enough info.4. Yes, ∆FHG~∆KHJ, AA~6. No, 20/45 ≠ 25/558. Yes, ∆NMP~∆NQR, SAS~10. AA~, x = 7.512. AA~, x = 12.833314. AA~, x = 816. SAS~, x = 12 m.18. AA~, x = 15 ft 9 in. 11. AA~, x = 2.517. AA~, x = 220 yd.
21. 151 m.24. Yes, ∆GMK~∆SMP, SAS~26. Yes, ∆XYZ~∆MNK, SSS~28. 45 ft30. 3:232. 12:734. 3:136. 3:237. 2:144. Any volunteers? ☺
∆ACB ~ ∆___ ~ ∆___using which postulate/theorem?
Pg 386 # 13, 15. Pg 400 # 2, 8-26 evens, 29, 32-33, 36-38 all, 40.
Pg 386: 13. AA~ Post, x = 12.15. AA~ Post, x = 15Pg 400: 2. x = 88. x = 7.5
10. x = 9.612. x = 4.814. x = 3.616. x = 1218. SQ (or QS)20. KP (or PK)
22. PM (or MP)24. LW (or WL)26. 671 ft29. x = 18 m, y = 12 m.32. 2.533. 2/7 or 3.
36. yes, since
37. no, because
38. yes, since
40. 4.5 cm or 12.5 cm
159
106=
1620
1215
=
1024
1228
≠
Proportional Parts in Triangles
E
Notes: Similarity in Right Triangles Name: ___________________________ Geometry 7-4 Date: ________________ Period: _____
Page 391: Hands-On Activity: Similarity in Right Triangles
1. Which angles have the same measure as ∠1? 2. Which angles have the same measure as ∠2? 3. Which angles have the same measure as ∠3? 4. Based on your results, what is true about the three angles?
5. Use the diagram at the right to complete the similarity statement.
∆RST ~ __________ ~ __________ Which segment is the altitude (perpendicular segment from a vertex) to the hypotenuse?
Page 392: Theorem 7-3: The altitude to the hypotenuse of a right triangle divides the triangle into two triangles that are __________ to the original triangle and to each other. 3-step process for using this theorem: a. Redraw all three triangles separately.
• Mark the right angle. • Distort so the medium leg is noticeably
longer than the short leg. • Label the vertices, lengths, angles, etc.
b. Make a chart. • Fill in the rows one
triangle at a time with both the known values and unknowns.
Small
leg Long leg
Hypo-tenuse
∆1
∆2
∆3
c. Set up a proportion that compares corresponding sides. Then solve!
hhr=
sb
b=
car
=
Class Exercises: Find the values of the variables. 15. a. Redraw all three triangles separately.
b. Make a chart. Small
leg Long leg
Hypo-tenuse
∆1
∆2
∆3
c. Set up a proportion. Solve for x.
17. a. Redraw all three triangles separately.
b. Make a chart. Small
leg Long leg
Hypo-tenuse
∆1
∆2
∆3
c. Set up a proportion. Solve for x.
Pg 394 # 6-22 evens, 21, 34, 36-38, 49-51.Pg 376 # 19-20.
Pg 394: 6. 258. 3 sqrt(7)
10. r12. a and a14. b16. 2018. 6 sqrt(3) ≈ 10.3920. 6022. ∆JKL ~ ∆KNL ~ ∆JNK21. a) 18 mi. b) 24 mi.
34. x = 12, y = 3 sqrt(7) ≈ 7.94, z = 4 sqrt(7) ≈ 10.58
36. x = 4, y = 2 sqrt(13) ≈ 7.21, z = 3 sqrt(13) ≈ 10.82
37. 12 sqrt(2)
38. C 49. 350. 4 51. 4.5Pg 376: 19. 70 mm.20. 54 in. by 87.37 in.
3
7
1313
77
2
Transformations: Pre-AP Extensions ☺ Name: ___________________________ Geometry chapter 9, especially section 9-6 Date: ________________ Period: _____ 1. The dashed figure is a translation image of the
solid figure. Write a rule to describe the translation. ____________________________
2. Draw the line of reflection you can use to map one figure onto the other. Find the equation of the line of reflection. ______________________________
3. Reflect DE first over the line y = –x to find '' ED
and then over the line y = –x – 6 to find '''' ED . Describe a single transformation (be specific) that
will map DE to '''' ED . D’(_____, _____) E’(_____, _____) D’’(_____, _____) E’’(_____, _____) Describe:
4. Reflect AB first over the line y = x – 1 to find
'' BA and then over the line x = –2 to find '''' BA . Describe a single transformation (be specific) that
will map AB to '''' BA . A’(_____, _____) B’(_____, _____) A’’(_____, _____) B’’(_____, _____) Describe:
5. Find the image of triangle FGH for a dilation with center P(-8, -1) and scale factor 3.
F’(_____, _____) G’(_____, _____) H’(_____, _____) Hint: What is the slope/distance of PF? What must be the slope/distance of PF’ ? of PG and PG’ ? of PH and PH’ ? 8. Find the image of
O(0, 0) after two reflections, first across l1 then across l2. l1: x = d l2: y = -2
O’’(_____, _____)
6. The composition of two reflections over parallel lines can be described as what single transformation? _______________ The composition of two reflections over intersecting lines can be described as what single transformation? _______________
7. Identify each mapping as a reflection, translation, rotation, or glide reflection. Also give the reflection line, translation rule, center & angle of rotation, or glide translation & reflection line.
a) ∆ABC → ∆EFC b) ∆CDE → ∆JIM c) ∆MJK → ∆ECD d) ∆ABC → ∆GHM e) ∆ABC → ∆MKJ
y y
x x
y y
D
E B xx
A
F P• G H
y
x
y
B D H I
G J A C E M
F K
x
More: Reflections and Compositions –
9. a) The graph shows quadrilateral TUVW, KN , and KL . At what coordinate point should vertex M be placed to make quadrilateral KLMN congruent to quadrilateral TUVW? b) What are the 2 lines of reflections that maps quadrilateral TUVW to quadrilateral KLMN ? c) Describe these 2 reflections as a single transform-ation (be specific).
10. a) Parallelogram WBMP is shown on the grid below. If WBMP is reflected across the line y = -x and then translated 4 units down to become parallelogram W B M P′ ′ ′ ′ , what will be the coordinates of M ′ ? b) What will be the coord. of M ′ if WBMP is translated first, then reflected? c) Find the line of reflection that will map W onto (-10, 5).
11. Reflect ∆ABC in the x-axis and then its image, ∆A'B'C' in the line y = -x. Label the coordinates of each triangle. a) What single rotation would map ∆ABC onto ∆A"B"C"? __________ In a different color, plot the images of ∆ABC after a 90° rotation, 180° rotation, and 270° rotation. Then write the rotations as a composition of two reflections: b) A 90° rotation is
a reflection across the line __________ then a reflection across __________ c) A 180° rotation is
a reflection across the __________ then a reflection across the __________ d) A 270° rotation is
a reflection across the line __________ then a reflection across __________ 12. A composition of transformations is two transformations where the second transformation is performed on the image of the first transformation. The symbology used for a composition of transformations is S T which can be read "S after T" or "T followed by S." Combining this notation with our arrow notation we can write S T: P P". This notation indicates that the composition of transformations maps P onto P". S and T are translations, R is a reflection, D is a dilation. Let S: (x, y) (x + 1, y + 4) T: (x, y) (x + 3, y - 1) R: (x, y) (-x, y) D: (x, y) (2x, 2y). Given A(4, 1), B(1, 5), and C(0, 1), find the coordinates of A", B", and C" under the following transformations (the order matters!): a) S T: A"________ B"________ C"________
T S: A"________ B"________ C"________ Is S T equal to T S ? b) S R: A"________ B"________ C"________
R S: A"________ B"________ C"________ Is S R equal to R S ? c) S D: A"________ B"________ C"________
D S: A"________ B"________ C"________ Is S D equal to D S ?