lesson 9.3 relating congruent and similar figures to...

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72 Chapter 9 Lesson 9.3

Lesson 9.3 Relating Congruent and Similar Figures to Geometric Transformations

State whether the figure and image are congruent or similar.

1. Rectangle ABCD is rotated 90° clockwise about vertex A.

2. A parallelogram is reflected in the x-axis and then reflected in the y-axis.

3. A photocopier dilates a picture by a scale factor of 34

.

4. A trapezoid is dilated with center (0, 0) and scale factor 21.

5. A hexagon is rotated 90° counter clockwise about its center (0, 0) and then dilated by a scale factor of 2.

6. ABC is mapped onto ABC under a transformation. ABC is the image of ABC under another transformation.

a) Describe the transformations that map ABC onto ABC and ABC onto ABC.

ABC is mapped onto ABC by using a reflection in the line . ABC is mapped onto ABC by using a rotation of about the point ( , ).

xC�

A� B�

C�

B�

A�

C

AB

y

1

0

2

3

4

2 3 41�1�1

�2

�3

�4

�2�3�4

MIF_ExtraPractice C3_Ch09.indd 72 3/30/12 12:34 AM

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Extra Practice Course 3B 73

b) If the order of the transformations is reversed, draw ABC and ABC and ABC on the coordinate plane.

c) Do the two triangles ABC have the same coordinates? Are they congruent? Explain.


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Solve.

7. A triangle ABC with vertices A (23, 4), B (23, 2) and C (26, 2) is reflected in the y-axis to obtain the image ABC. ABC then is mapped onto ABC shown in the diagram by another transformation.

x

y

1�1�2�3�4�5�6 2 3 4 5 6

1

2

3

4

5

6

0

�1

�2

�3

�4

�5

�6

B

A

C

B� A�

C�

a) Draw ABC on the same axes above.

b) Describe the transformation that maps ABC onto ABC .

c) Describe a single transformation that maps ABC onto ABC.


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Solve.

8. A triangle PQR with vertices P (22, 2), Q (21, 3) and R (21, 1) is dilated by a scale factor 2 with center P to obtain the image PQR. PQR is then mapped by another transformation onto PQR shown in the diagram.

x

y

1�1�2�3 2 3 4 5 6 7

1

2

3

4

5

6

0

�1

�2

R

Q

P

Q �

P �

R �

a) Draw PQR on the same axes above.

b) Describe the transformation that maps PQR on PQR.

c) Describe a single transformation that maps PQR onto PQR.


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Solve.

9. ABC is mapped onto ABC under a transformation. ABC is then mapped onto ABC under another transformation. Describe the sequence of transformations from ABC to ABC.

a)

x

y

1�1�2�3�4 2 3 4

1

2

3

4

0

�1

�2

�3

�4

B

A� B�

C�A�

B �C �

C

A


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Solve.

b)

x

y

1�1�2�3�4�5 2 3 4 5 6 7 8

1

2

3

4

5

0

�1

�2

�3

�4

�5

�6

�7

�8

A

BC

B �

A�

C �

A�

C�B�


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Solve.

c)

x

y

1�1�2�3�4�5�6�7 2 3 4 5 6 7

1

2

3

4

5

0

�1

�2

�3

�4

�5

�6

�7

A

BC

A�

C�

C� B �

A�

B�


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Solve.

10. Quadrilateral ABCD is dilated with center C and scale factor 1.5. It is mapped onto PQRS. The length of AB is 3 feet and the area of ABCD is 12 square feet.

a) Find mQRS.

b) Find the length of PQ.

c) Determine the area of PQRS.

11. The area of a rectangular postcard is 60 square centimeters. A dilated copy has an area of 240 square centimeters. By what scale factor is the diagonal of the postcard enlarged?

A

D

C

120°

3 ft

Area = 12 ft2

80°

B



�

PXBX

PABQ

�

PX10

2412

Length of PX � 2412

? 10

5 20 ft Length of the pole BP 5 10 1 20 5 30 ft

In PAB, AB 5 �30 242 2

5 18 ft The distance between the walls is 18 feet. In ABQ, AQ 5 �18 122 2

5 21.63 21.6 ft The length of pole AQ is 21.6 feet.

19. a) 1st pair: AOB DOC Explanation: mAOB 5 mDO C (vertically opposite s) mABO 5 mDC O (alternate s, AB CD) AOB DOC (equiangular) 2nd pair: OCD OEF Explanation: mCOD 5 mEOF

(common ) mOCD 5 mOEF

(corresponding s, CD EF) OCD OEF (equiangular) 3rd pair: AOB FOE Explanation: mAOB 5 mFOE (vertically

opposite s) mABO 5 mFEO (alternate

s, AB EF) AOB FEO (equiangular)

b) AOE is not similar to BOF. Explanation: Only one pair of angles

(AOE and BOF) are equal. The remaining two pairs are not equal.

AOE is not similar to BOF.

Lesson 9.3 1. Congruent 2. Congruent 3. Similar 4. Congruent 5. Similar

6. a) y 5 x; 90° clockwise; 0; 0.

b)

0x

2 3�2 �1 1

2

3

1

4

A″

C″C

B″

C′

A′

B′

BA

�3 4

y

c) No; the coordinates are different because the order of transformations is reversed. Yes; the size and shape of figures remain the same under translations and reflections.

7. a)

0x

2 3�2 �1 1

2

3

1

4

5A A�

C�

B�BC

6

�4 �3�5�6 4 5 6

y

b) A’B’C is mapped onto A”B”C” by

a rotation of 90° clockwise about origin O.

c) A reflection about the line y 5 x will map

ABC onto A”B”C”.

8. a)

x

y

1�1 0�2�3 2 3 4 5 6 7

2

1

�1

�2

3

4

5

6

Q′ Q″

P″

R″

P′

R′R

P

Q

MIF_ExtraPractice_C3_Ch07-11_Ans.indd 181 30/03/12 7:09 AM

182 Answers

b) P’Q’R is mapped onto P”Q”R” by a reflection about the line x = 2.5.

c) A dilation of scale factor 22 about the point (1, 2) will map PQR to P”Q”R”.

9. a) ABC is mapped onto A’B’C’ by a translation of 4 units to the right and 6 units up.

A’B’C’ is mapped onto AB C by a rotation of 90° clockwise about the point (0, 2).

b) ABC is mapped onto A’B’C’ by a reflection about the line x 5 1. A’B’C’ is mapped onto A”B”C” by a dilation of scale factor 23 about the point (21, 1).

c) ABC is mapped onto A’B’C’ by a rotation of 90° counter clockwise about the point (22, 0). A’B’C’ is mapped onto A”B”C” by a reflection about the line y 5 x.

10. ABCD PQRS a) mQRS 5 mBCD 5 360° 2 mDAB 2 mABC

2 mCDA 5 360° 2 90° 2 120° 2 80° 5 70° (sum of quadrilateral)

b) � 1.5PQAB

(scale factor)

� 1.5PQ

3

PQ 5 3 ? 1.5 5 4.5 ft

c) � (1.5)PQRSABCD

Area ofArea of

2

� (1.5)PQRSArea of12

2

Area of PQRS 5 (1.5)2 ? 12 5 27 ft2

The area of PQRS is 27 square feet.

11. Let the scale factor be k.

� k

Area of enlarged copyArea of originalpostcard

2

� k240

602

4 5 k2

k 5 2

The diagonal of the postcard is dilated by a scale factor of 2.

Brain@Work1. Let the height of the big triangle be p.

� �

pm4

34

4p 5 3(4 1 m)

4p 5 12 1 3m –– Eq. 1

p

n m435 � �

�

5p 5 3(n 1 4 1 m)

5p 5 3n 1 12 1 3m

5p 5 3m + 3n 1 12 –– Eq. 2

Multiply Eq. 1 by 5:

5 4p 5 5 (12 1 3m)

20p 5 60 1 15m –– Eq. 3

Multiply Eq. 2 by 4:

4 5p 5 4 (3m 1 3n 1 12)

20p 5 12m 1 12n 1 48 –– Eq. 4

Substitute Eq. 3 into Eq. 4:

60 1 15m 5 12m 1 12n 1 48

60 1 15m 2 60 5 12m 1 12n 1 48 – 60

15m 5 12m 1 12n 2 12

15m 2 12m 5 12m 1 12n 2 12 2 12m 3m 5 12n 2 12

�

�m n3 12 1233

m 5 4n – 4

2. XAY 5 BAZ [common angle]

AYX 5 AZB [corr. s]

XAY and BAZ are similar.

�

�

AYAZ

XYBZ

AZBZ

AYXY

XBW 5 ABZ [common angle]

BWX 5 BZA [corr. s]

XBW and ABZ are similar.

�

�

BWBZ

XWAZ

AZBZ

XWBW

�So, AY

XYXWBW .

Since XW 5 YZ and WZ 5 XY,

�

�

AYWZ

YZBW

AYYZ

ZWWV

So, AY : YZ 5 ZW : WB (shown)

Cumulative Practice for Chapters 7 to 9 1. p2 5 262 1 242

MIF_ExtraPractice_C3_Ch07-11_Ans.indd 182 30/03/12 7:09 AM

lesson 9.3 relating congruent and similar figures to...

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