Download Foundations 20 Unit 1 - Logic Puzzles

Math 20 Foundations Unit 1 Deductive, Inductive Reasoning & Puzzles Proofs
Assignment #1
1. The distance from Regina to Saskatoon is approximately 250 km. If road conditions are such that a motorist
can only travel 50 km/h on the trip to Saskatoon, at what speed must the motorist drive on the return trip in
order to average 100 km/h?
2. If it were possible to wrap the earth with a metal ring at its equator, you would need a ring whose
circumference was approximately 40 000 km. If you inserted an extra 2m (0.002 km) into the ring so that it is
now 40 000.002 km in length and no longer snug against the earth, do you believe there is enough room to
crawl between the earth and the ring? Check your answer. (Hint: C = 2  r )
3. Three fuel savings devices were invented in the same year. The first claimed to reduce fuel consumption by
10%, the second by 40% and the third by 50%. If a motorist attached all three devices to his/her motor, how
much was his/her fuel savings?
Reasoning Puzzles Assignment #2
#1 The Baseball Team
The problem is: Who plays which position?
Clues:
1. Smith and Brown each won $10 playing cards with the pitcher.
2. Hunter is taller than Knight but shorter than White and all weigh more than the first baseman.
3. The third baseman used to live across the corridor from Jones in an apartment block.
4. Miller and the outfielders play bridge.
5. White, Miller, Brown, the right fielder, and centre fielder are single, the rest are married.
6. One of Adams or Knight plays in the outfield.
7. The right fielder is shorter than the centre fielder.
8. The third baseman is the brother of the pitcher’s wife.
9. Green is taller than the infielders, the pitcher, and the catcher, except for Jones, Smith, and Adams.
10. The second baseman beat Jones, Brown, Hunter, and the catcher at cards.
11. The third baseman, shortstop, and Hunter made $150 on the stock market.
12. The second baseman was engaged to Miller’s sister.
13. Adams lives in the same house as his sister, but does not like the catcher.
14. Adams, Brown, and the shortstop lost $200 each on the stock market.
15. The catcher has three daughters.
16. The third baseman has two sons.
17. Green is being sued for divorce.
Catcher
1
2
3
4
5
6
7
8
9
Smith
Brown
Hunter
Knight
White
Jones
Miller
Adams
Green
Pitcher
1st
2nd
3rd
baseman baseman baseman
Short
Stop
Right
Fielder
Centre
Fielder
Left
Fielder
#2 Who is the Baker?
a) Who is the Baker? _______________
c) Who is going to BARKER STREET?
b) What is Burt’s last name? _______________
_______________
Clues:
1. BRAD is married to BETTY.
2. BARBARA’S husband gets into the 3rd cab.
3. BART is a BANKER.
4. The last cab goes to BARTON STREET.
5. BEATRICE lives on BURTON STREET.
6. The BUTCHER gets into the 4th cab.
7. BRETT gets into the 2nd cab.
8. BERNICE is married to the BROKER.
9. Mr. BARKER lives on BURTON STREET.
10. Mr. BURGER gets into the cab before (in front of) BRENDA’S husband.
11. Mr. BUNGER gets into the 1st cab.
12. Mr. BAKER lives on BOURBON STREET.
13. The BARBER lives on BAKER STREET.
14. Mr. BAKER gets into the cab before (in front of) Mr. BURKE.
15. The BARBER is three cabs before (in front of) BRIAN.
16. Mr. BURGER is in the cab before the BUTCHER.
Cab Number
1
2
3
4
5
First Name
Last Name
Profession
Wife’s Name
Destination
#3 Golf
A bricklayer, plasterer, carpenter, tinsmith, and roofer enjoy playing golf. Their names in no particular order
are Alex, Ted, John, Brian and Mark. They played a round of golf with the following results:
 Alex scored a par of 72, beating the roofer by two strokes.
 Brian finished ten strokes over par, twelve strokes more than the bricklayer.
 The tinsmith beat Mark by four strokes and won the round.
 John beat the carpenter by eight strokes, but did not win the round.
The problem is: What is each person’s profession and golf score?
Score
Alex
Ted
John
Brian
Mark
Bricklayer
Plasterer
Carpenter
Tinsmith
Roofer
#4 Einstein Quiz
There are 5 houses in 5 different colours. In each house lives a person with a different nationality. These 5
owners drink a certain beverage, smoke a certain brand of cigar, and keep a certain pet. No owners have the
same pet, smoke the same brand of cigar or drink the same drink.
The problem is: Who keeps the fish?
Clues:
1. The Brit lives in a red house.
2. The Swede keeps dogs as pets.
3. The Dane drinks tea.
4. The green house is on the left of the white house.
5. The green house owner drinks coffee.
6. The person who smokes Pall Mall rears birds.
7. The owner of the yellow house smokes Dunhill.
8. The man living in the house right in the centre drinks milk.
9. The Norwegian lives in the first house.
10. The man who smokes Blend lives next to the one who keeps cats.
11. The man who keeps horses lives next to the man who smokes Dunhill.
12. The owner who smokes Blue Master drinks beer.
13. The German smokes Prince.
14. The Norwegian lives next door to the blue house.
15. The man who smokes Blend has a neighbour who drinks water.
Nationality
Colour
Drink
Pet
Smoke
Co-ordinate Geometry Proofs Assignment #2
1. Supply the missing coordinates without using any new letters.
a)
POST is a square
b)
 MON is isosceles
c)
JOKL is a trapezoid
d)
GOLD is a rectangle
d)
Right  TOP is isosceles
f)
Rectangle ABCD
g)
Parallelgram ABCD
h)
Square ABCD
i)
Isosceles  ABC
j)
Parallelogram ABCD
k)
Isosceles Trapezoid ABCD
l)
Equilateral  ABC
2. Use coordinate geometry to prove the following.
a) If a point lies on the perpendicular bisector of a segment, then the point is equidistant from the endpoints
of the segment.
b) The diagonals of a rectangle are congruent.
c) The diagonals of a square are perpendicular.
d) The diagonals of an isosceles trapezoid are congruent.
e) The medians drawn to the legs of an isosceles triangle are congruent.
f) The quadrilateral formed by joining the midpoints of the sides of a rectangle is a rhombus.
g) The segment joining the midpoints of two sides of a triangle is equal to one half the length of the third
side.
h) The segment joining the midpoints of two sides of a triangle is parallel to the third side.
i) The medians of an equilateral triangle all have the same length.
j) The median of an isosceles triangle is also an altitude.
Inductive Reasoning Proofs Assignment #3
Prove the following using mathematical induction.
a) 1  3  5  ...  2n  1  n 2
b) 2  4  6  ...  2n  nn  1
c) 1  2  4  ...  2 n1  2 n  1
1 1 1
1
1
d) 1     ...  n  2  n1
2 4 8
2
2
1
e) 1  2  3  ...  n  nn  1
2
2
f)
g)
h)
i)
1

1  2  3  ...  n   nn  1
2

1
12  23  34  ...  nn  1  nn  1n  2
3
1
13  24  35  ...  nn  2  nn  12n  7 
6
1
123  234  345  ...  nn  1n  2  nn  1n  2n  3
4
3
3
3
3
Math 30C Unit VI Mathematical Proofs
Number Proofs Assignment #4
Name the property that justifies each step in the following.
1. 2 x (571 x 5)
2 x (5 x 571)
(2 x 5) x 571
10 x 571
5710
_________________
_________________
_________________
_________________
2. (-1 + c) + 1
[c + (-1)] + 1
c + [(-1) + 1]
c+0
c
_________________
_________________
_________________
_________________
3. 27 + 3(h + 1)
27 + (3h + 3)
(3h + 3) + 27
3h + (3 + 27)
3h + 30
_________________
_________________
_________________
_________________
4. 27 + 3(h + 1)
27 + (3h + 3)
(3h + 3) + 27
3h + (3 + 27)
3h + 30
_________________
_________________
_________________
_________________
5. 4(2t – 5) + 7
8t – 20 + 7
8t – 13
8t – 13 + 13
8t
8t – 3t
5t
t
6.
y 7y

4 3
 y 7y 
 
12
4 3 
3y – 28y
-25y
y
=
=
=
=
=
=
=
=
3t – 58
3t – 58
3t – 58
3t – 58 + 13
3t – 45
3t – 45 – 3t
-45
-9
_________________
_________________
_________________
_________________
_________________
_________________
_________________
=
5
_________________
=
5(12)
_________________
=
=
60
60
 60  12

25
5
_________________
_________________
=
_________________
Deductive Reasoning Statements
1. Given.
2. Definition of Perpendicular Lines – two lines that form a right angle.
3. Definition of Right Angle – an angle of 90° is formed.


4. Definition of Angle Bisector – the bisector of  ABC is a BD in the interior of  ABC such that
 ABD   DBC.
A
D
B
C
5. Definition of Segment Bisector – a line, ray, segment, or plane that intersects the segment at its
midpoint.
6. Definition of Perpendicular Bisector – a line, ray, segment, or plane that is a segment bisector and is
perpendicular to the segment at its midpoint.
7. Definition of Altitude of a Triangle – a segment starting from a vertex that is perpendicular to the side
opposite that angle.
8. Definition of Median of a Triangle – a segment starting from a vertex to the midpoint of the side
opposite that angle.
9. Definition of Midpoint of a Segment – the point that divides the segment into two congruent segments.
10. Definition of Complementary Angles – two angle whose measures have a sum of 90°.
 ABD +  DBC = 90°
A
D
B
C
11. Definition of Supplementary Angles – two angle whose measures have a sum of 180°.
D
 ABD +  DBC =180°
A
B
C
12. Properties of Equality:
a. Addition Property
If a = b and c = d, then a + c = b + d
b. Subtraction Property
If a = b and c = d, then a – c = b – d
c. Multiplication Property
If a = b, then ca = cb.
d. Division Property
a b
If a = b and c  0, then  .
c c
e. Symmetric Property
If a = b, then b = a.
f. Transitive Property
If a = b and b = c, then a = c.
g. Substitution Property
If a = b, then either a or b may be substituted for the other in any equation or equality.
13. BAIT – Base Angles of an Isosceles Triangle are Congruent.
A
If AB  AC , then  ABC   ACB
B
C
14. The sum of the measures of the interior angles of a triangle is 180°.
15. Vertically Opposite angles are congruent.
16. Properties of Parallel Lines.
If two parallel lines are cut by a transversal, then:
a. Alternate interior angles are congruent.
b. Corresponding angles are congruent.
c. Same side interior angles are supplementary.
17. Triangle Congruency Methods
a. SSS – side, side, side.
b. SAS – side, angle, side.
c. ASA – angle, side, angle.
d. AAS – angle, angle, side
e. HL – (right triangles only) hypotenuse, leg
18. The radius of a circle and a tangent drawn to it at the point of tangency are perpendicular to each other.
OP  PT where point P is the point of
Tangency.
O
P
T
19. The tangents drawn to a circle from the same point outside the circle are congruent.
R
O
S
RS  TS
T
20. If a chord of a circle is bisected by the diameter of that circle, the chord is perpendicular to the diameter.
If AD  BD
then AB  OD
O
A
D
B
21. If two chords intersect inside a circle, the product of the segments of one chord equals the product of the
segments of the other chord.
a
c
d
b
a∙b = c∙d
Deductive Proofs Assignment #5
1. Given: Parallelogram ABCD
M and N are midpoints of AB and DC.
D
N
C
Prove: AMCN is a parallelogram.
A
M
B
D
N
C
2. Given: Parallelogram ABCD
AN and CM bisect A and C.
Prove: AMCN is a parallelogram.
A
M
3. Given: Parallelogram ABCD
W, X, Y, and Z are midpoints of AO, BO, CO, DO.
B
D
C
Z
Y
W
Prove: WXYZ is a parallelogram.
O
X
A
B
D
4. Given: Parallelogram ABCD
DE  BF
C
E
F
Prove: AFCE is a parallelogram.
A
B
E
D
5. Given: AE  CD, DBC  C, and A  DBC.
Prove: ABDE is a parallelogram.
A
B
6. Given: Parallelograms ABCD and CDFE.
C
D
Prove: ABEF is a parallelogram.
A
C
F
B
7. Given: L is the midpoint of KM.
JK MN
E
J
K
L
Prove: L is the midpoint of JN.
M
N
E
8. Given:  1   2
AB  CD
Prove:  3   4
A
B
C
D
G
9. Given:  F   H
GI bisects  FGH
F
H
Prove: FI  HI
I
10. Given:  ADC and  CBA are right angles
DC  AB
D
C
E
Prove: AE  CE
A
B
N
34
11. Given: MN  ON
1 2
Prove:  3   4
P
1
M
2
N
J
K
12. Given: JKLM is an isosceles trapezoid
JM  KL; 1  2
Prove: JKNM is a parallelogram.
M
N
L
A
13. Given: PA and PB are tangent to circle O
Prove:  1   2
O
P
B
14. Given: Refer to diagram to the right
9
x
Prove: x = 12
16
x
C
15. Given: Tangent circles O and Q with tangents
PA, PB, PC.
Prove: PA  PC
Q
B
O
P
A
16. Given: Chord PQ is bisected by
P
diameter RS of circle O.
1  2
Prove:  PTS   QTS
R
T
S
Q
U
17. Given: UV is tangent to circles R and S at Q.
RU  SU
R
Q
S
Prove: RV  SV
V
18. Given: Refer to diagram to the right
Prove: x = 2 or 6
4
8–x
3
x
19. Given: JT is tangent to circle O
JO  13; OT  5
O
Prove: JT  12
T
J
20. Given: JT is tangent to circle O
JK  2; KO  3
O
Prove: JT  4
K
T
J
Indirect Proofs Assignment #6
1. Given: XYZ; m  X  100 °
Z
Prove:  Y is an acute  .
X
Y
t
2. Given: Transversal t passes through lines a and b.
m 1 m2
1
a
Prove: a b
3
b
2
J
3. Given: OJ  OK; JE  KE
1


Prove: OE doesn't bisect  JOK.
E
2
O
K
4. Given:  1   2; OJ  OK
use question #3 diagram.
Prove:  J and  K are not both right angles.
V
5. Given:  RVT and  SVT are equilateral.
 RVS is not equilateral.
Prove:  RST is not equilateral.
R
S
T
6. Given: AT  BT  5; CT  4
A
5
Prove:  ABC is not a right  .
T
4
5
C
7. Given: Coplanar lines l, m, and n.
n intersects l at point P;
l m
Prove: n intersects m
B
l
m
8. Prove that the diagonals of a trapezoid do not bisect each other.
9. Prove that no regular polygon has a 155° angle.
P
n