Geometry Flexbook Q1 Solutions – Chapter 1
Chapter 1: Geometric Transformations
1.1 Basic Geometric Definitions
1. sample answer:
D
A
B
P
C
2. sample answer:
E
B
A
C
D
3. sample answer:
Y
X
Z
W
P
4. sample answer:
G
Q
H
P
5. sample answer:
I
J
K
L
O
⃡ , 𝑊𝑌
⃡ , 𝑋𝑌
⃡ , 𝑌𝑋
⃡
6. sample answer: line m, 𝑊𝑋
⃡
7. sample answer: 𝑇𝑆, 𝑇𝑅 , 𝑅𝑇
8. sample answer: plane V, plane RST
9. sample answer: A plane is the best geometric model for a soccer field because it is a flat surface.
10. sample answer: sun rays, lasers
11. If a line intersects a plane that does not contain the line, then it intersects the plane in exactly one
point.
12. A postulate is a basic rule of geometry that is assumed to be true. A theorem is not assumed to be
true, it is proven.
13. sample answer: ⃡𝑅𝑆 and 𝑃𝑄 intersect at point Q. R, Q, and S are collinear. P and Q are collinear.
̅̅̅̅ intersect at point
14. sample answer: A, B, and C are coplanar. Point D is not on plane ABC. 𝐴𝐶 and 𝐴𝐵
A.
15. sample answer: Plane J contains points E and H. 𝐸𝐹 intersects plane J at point E. 𝐻𝐺 intersects
plane J at point H.
16. sample answer: Rays IM, IL, IK, and IJ all intersect at point I.
17. true
18. false
19. false
20. true
21. false
22. true
23. false
24. false
25. true
1.2 Angle Classification
1. straight 2. obtuse 3. acute 4. acute 5. right 6. obtuse 7. right 8. obtuse 9. acute 10. obtuse
11. right 12. straight 13. acute 14. acute 15. obtuse
1.3 Parallel and Perpendicular Line
1. sample answer: ̅̅̅̅
𝑋𝑌 and ̅̅̅̅
𝐴𝑉, ̅̅̅̅
𝑋𝑌 and ̅̅̅̅
𝐸𝑍
̅̅̅̅
̅̅̅̅
2. sample answer: 𝑉𝑍 and 𝐴𝐸
3. sample answer: ̅̅̅̅
𝐷𝑌 and ̅̅̅̅
𝑋𝑌
̅̅̅̅
4. one; 𝐴𝑉
̅̅̅̅
5. one; 𝐷𝐶
6. sample answer: the ocean and the horizon
7. sample answer: the inside of a mineral crystal
8. sample answer: snowflakes
1.4 Identify and Use the Distance Formula
1. 5 2. 13 3. √128 ≈ 11.3 4. 5 5. √26 ≈ 5.1 6. 6 7. √41 ≈ 6.4 8. 8 9. √29 ≈ 5.4 10. √10 ≈ 3.2
11. √145 ≈ 12𝑘𝑚 12. 5km 13. 4 km 14. 6 km 15. √73 ≈ 8.5 𝑘𝑚
1.5 Geometric Translations
1. 𝐴′ (4, −6) 2. 𝐵′(7, −4) 3. 𝐶′(9, −11) 4. 𝐴"(9, −15) 5. 𝐷(7, 16) 6. 𝐴‴ (14, −24) 7. The four
9
points are collinear with a slope of − 5 8. 𝐴′ (−8, −14)𝐵′ (−5, −17)𝐶′(−7, −5)
9. 𝐴′ (5, −3)𝐵′ (8, −6)𝐶′(6, 6) 10. 𝐴′ (−6, −10)𝐵′ (−3, −13)𝐶′(−5, −1)
11. 𝐴′ (−11, 1)𝐵′ (−8, 2)𝐶′(−10, 10) 12. 𝐴′ (−5, −7)𝐵′ (−2, −10)𝐶′(−4, 2)
13. 𝐴′ (−3, 3)𝐵′ (0, 0)𝐶 ′ (−2, 12) 14. (𝑥, 𝑦) → (𝑥 − 6, 𝑦 + 2) 15. (𝑥, 𝑦) → (𝑥 + 9, 𝑦 − 7)
̅̅̅̅ = √26 𝐵𝐶
̅̅̅̅ = √13 𝐶𝐴
̅̅̅̅ = √13
16. (𝑥, 𝑦) → (𝑥 − 3, 𝑦 − 5) 17. (𝑥, 𝑦) → (𝑥 + 8, 𝑦 + 4) 18. 𝐴𝐵
̅̅̅̅ = √26 𝐵𝐶
̅̅̅̅ = √13 𝐶𝐴
̅̅̅̅ = √13 20. ∆ABC and ∆A’B’C’ are congruent. Any translation creates a
19. 𝐴𝐵
congruent image because a translation is a rigid transformation that preserves distance and angle
measure. 21. (𝑥, 𝑦) → (𝑥 + 6, 𝑦 − 2) 22. (𝑥, 𝑦) → (𝑥 − 9, 𝑦 + 7) 23. (𝑥, 𝑦) → (𝑥 − 6, 𝑦 − 4)
24. (𝑥, 𝑦) → (𝑥 + 5, 𝑦 + 11)
25. (𝑥, 𝑦) → (𝑥 − 3, 𝑦 − 7)
1.6 Rules for Rotations
Starting point
1.
(1, 4)
2.
(4, 2)
3.
(2, 0)
4.
(-1, 2)
5.
(-2, -3)
6. R180°(x, y)→(-x, -y)
7. R270°(x, y)→(y, -x)
8. R90°(x, y)→(-y, x)
9. R270°(x, y)→(y, -x)
10. R180°(x, y)→(-x, -y)
11. R90°(x, y)→(-y, x)
90° rotation
(-4, 1)
(-2, 4)
(0, 2)
(-2, -1)
(3, -2)
180° rotation
(-1, -4)
(-4, -2)
(-2, 0)
(1, -2)
(2, 3)
270° rotation
(4, -1)
(2, -4)
(0, -2)
(2, 1)
(-3, 2)
360° rotation
(1, 4)
(4, 2)
(2, 0)
(-1, 2)
(-2, -3)
12. 9. R270°(x, y)→(y, -x)
13-15 are not necessary for this course.
1.7 Reflections
Each of the points were assigned letters to make it easier to read the graphs.
1. A’(3, -1) 2. B’(4, 2) 3. C’(-5, -3) 4. D’(-6, -4)
5. E’(-4, 3) 6. F’(-5, 4) 7. G’(5, -4) 8. H’(-3, -3)
9. I’(1, 3) 10. J’(-2, 4) 11. K’(3, -5) 12. L’(4, -6)
13. reflection across the x-axis
14. reflection across the y-axis
15. reflection across the line y=x
1.8 Composition of Transformations
1. sample answer: An isometry, or rigid transformation, does not change the size or shape of a figure.
Even if isometries are combined, each image will still be the same size and shape so the new images that
result must also be isometries.
2. a translation of 2h units where h is the number of units the original parallel lines were apart
3. a rotation of 180 degrees about the origin
4. (2, -2) (-2, -4) (0, -8) (4, -6)
5. (x, y) →(x+6, -y)
6. (x, y)→(x-6, -y)
7. (2, -2) (-2, -4) (0, -8) (4, -6)
8. (x, y)→(x+6, -y)
9. The rules in #8 and #5 are the same. In #5, the y-coordinate is changed first with a reflection and in
#8, the x-coordinate is changed first with a translation but when you create the composition of the
reflection and translation, you have the same rule.
10. (-2, -3) (-9, -3) (-4, 2)
11. (x, y) →(-x, y-5)
12. (x, y) →(-x, y+5)
13.
14. a translation down 12 units
15. (x, y) →(x, y-12)
16.
17. a translation up 12 units
18. (x, y) →(x, y+12)
19. The triangles are congruent; however, one triangle is 12 units lower than the preimage and on
triangle is 12 units higher than the preimage.
20.
21.
22. The trapezoids in #20 & #21 are the same because the composition of both transformations is
(x, y)→(-x,-y). This composition is the same as a 180 degree rotation of the original image.
23. 14 units apart
24. 14 units
25. 330°
26. The center of rotation is the point of intersection the two lines of reflection.
27. 166°
28. 122°
29. 49°
30. 31 units