© T Madas. 20 cm 30 cm 40 cm A cuboidal box measures 30 cm by 40 cm by 20 cm. What is the longest...
-
Upload
felicity-amberlynn-harmon -
Category
Documents
-
view
221 -
download
0
Transcript of © T Madas. 20 cm 30 cm 40 cm A cuboidal box measures 30 cm by 40 cm by 20 cm. What is the longest...
© T Madas
© T Madas
20 c
m
30 cm 40 cm
A cuboidal box measures 30 cm by 40 cm by 20 cm.
What is the longest stick which fits in this box?
It should be obvious that the stick must be placed in a “diagonal” fashion
Is this the longest?
How do we find this distance?
© T Madas
30 cm 40 cm
20 c
m
A cuboidal box measures 30 cm by 40 cm by 20 cm.
x
Using Pythagoras:
30 2 + 40
2 = x 2
900 + 1600x 2 =
2500x 2 =
2500x =
50x =
Note: strictly ±50, the minus rejected since it is a length
50 cm
d
What is the longest stick which fits in this box?
© T Madas
30 cm 40 cm
20 c
m
A cuboidal box measures 30 cm by 40 cm by 20 cm.
What is the longest stick which fits in this box?
Using Pythagoras:
50 2 + 20
2 = d 2
2500 + 400d 2 =
2900d 2 =
2900d =
53.9 cmd ≈50 cm
d
© T Madas
30 cm 40 cm
20 c
m
A cuboidal box measures 30 cm by 40 cm by 20 cm.
What is the longest stick which fits in this box?
53.9 cm
© T Madas
( )30 cm 40 cm
20 c
m
A cuboidal box measures 30 cm by 40 cm by 20 cm.
53.9 cm
What is the angle between the stick and one of the 30 x 40 faces?
50 cm
Using Trigonometry:
2050
= tanθ
θ
tanθ = 25
θ = 25
tan-1
θ ≈ 21.8°
What is the longest stick which fits in this box?
© T Madas
© T Madas
© T Madas
[x ≈ 5.66 cm]
8 cm
8 cm
A pyramid has a height of 20 cm and a square base with side length of 8 cm.
20 c
m
1. Calculate the length of one of its sloping edges
4 cm
4 c
m
Using Pythagoras:
4 2 + 4
2 = x 2
16 + 16x 2 =
32x 2 =
32x =
x
32
© T Madas
+ 32
y ≈ 20.8 cm
8 cm
8 cm
A pyramid has a height of 20 cm and a square base with side length of 8 cm.
20 c
m
1. Calculate the length of one of its sloping edges
20 c
m
Using Pythagoras:
20 2 = y
2
400 + 32y 2 =
432y 2 =
432y =
y
32
32
2
© T Madas
8 cm
8 cm
A pyramid has a height of 20 cm and a square base with side length of 8 cm.
20 c
m
1. Calculate the length of one of its sloping edges2. Find the angle between one of its sloping faces and
its base.
θ4
(5)
Using Trigonometry:
204
= tanθ
tanθ =
θ = tan-1
θ ≈ 78.7°
5
© T Madas
2. Find the angle between one of its sloping faces and its base.
8 cm
8 cm
A pyramid has a height of 20 cm and a square base with side length of 8 cm.
20 c
m
1. Calculate the length of one of its sloping edges
θ4
3. Calculate the total surface area of the pyramid
d
[x ≈ 20.4 cm]
Using Pythagoras:
4 2 + 20
2 = d 2
16+ 400d 2 =
416d 2 =
416d =
416
© T Madas
2. Find the angle between one of its sloping faces and its base.
8 cm
8 cm
A pyramid has a height of 20 cm and a square base with side length of 8 cm.
20 c
m
1. Calculate the length of one of its sloping edges
4
3. Calculate the total surface area of the pyramid
S = x 8 x 4x 416
416
12
x 8+ 8
= 16 x 416 + 64
= 16 x 16 x 26 + 64
= 16 x 4 26 + 64
= 64 26 + 64
= 64[ ]26 + 1 ≈ 390 cm2
© T Madas
© T Madas
The picture below shows a triangular prism.
ABC is a right angle, BC = 5 cm, ABED is a square with a side length of 12 cm and M is the midpoint of AD.Calculate:1. the length AC
A
B
C
D
E
F
M
5
126
6All lengths in cm
A B
C
5
12
AC 2 =52+ 122
AC 2 =25+ 144
AC 2 =169
AC = 169AC = 13 cm
13
© T Madas
The picture below shows a triangular prism.
ABC is a right angle, BC = 5 cm, ABED is a square with a side length of 12 cm and M is the midpoint of AD.Calculate:2. the length BD
A
B
C
D
E
F
M
5
126
6All lengths in cm
D A
B
12
12
BD 2 =122+ 122
BD 2 =144+ 144
BD 2 =288
BD = 288BD ≈16.97 cm
13288
© T Madas
The picture below shows a triangular prism.
ABC is a right angle, BC = 5 cm, ABED is a square with a side length of 12 cm and M is the midpoint of AD.Calculate:3. the length CD
A
B
C
D
E
F
M
5
126
6All lengths in cm
D B
C
5
288
CD 2 =52+ 288 2
CD 2 =25+ 288
CD 2 =313
CD = 313CD ≈17.69 cm
13288
313
© T Madas
The picture below shows a triangular prism.
ABC is a right angle, BC = 5 cm, ABED is a square with a side length of 12 cm and M is the midpoint of AD.Calculate:4. CAB
A
B
C
D
E
F
M
5
126
6All lengths in cm
13288
313
A B
C
5
12θ
tanθ = 512
θ = 512
tan-1
θ ≈22.6°
© T Madas
The picture below shows a triangular prism.
ABC is a right angle, BC = 5 cm, ABED is a square with a side length of 12 cm and M is the midpoint of AD.Calculate:5. CDB
A
B
C
D
E
F
M
5
126
6All lengths in cm
13288
313
D B
C
5φ
tanφ =5
288
φ =5
288
tan-1
φ ≈16.4°
288
313
© T Madas
The picture below shows a triangular prism.
ABC is a right angle, BC = 5 cm, ABED is a square with a side length of 12 cm and M is the midpoint of AD.Calculate:6. CMB
A
B
C
D
E
F
M
5
126
6All lengths in cm
M A
B
12
6
BM 2 =62+ 122
BM 2 =36+ 144
BM 2 =180
BM =180BM ≈13.42 cm
180
© T Madas
The picture below shows a triangular prism.
ABC is a right angle, BC = 5 cm, ABED is a square with a side length of 12 cm and M is the midpoint of AD.Calculate:6. CMB
A
C
D
E
F
M
5
126
6All lengths in cm
B
M B
C
5α
tanα =5
180
α =5
180
tan-1
α ≈20.4°
180
180
© T Madas
© T Madas
x
y
z
A
BC
DE
FG
O
(6,3,2)
(6,3,0)
A cuboid OABCDEFG 6 units long by 3 units wide by 2 units high is drawn on a set of 3 dimensional axes as shown below.
1. Write down the coordinates of points B and F.
2. Calculate the length of OF.
6
3
2
© T Madas
x
y
z
A
DE
G
O
A cuboid OABCDEFG 6 units long by 3 units wide by 2 units high is drawn on a set of 3 dimensional axes as shown below.
1. Write down the coordinates of points B and F.
2. Calculate the length of OF.
6
3
2
C
O A
B
3
6
OB 2 = 62+ 32
OB 2 =36+ 9
OB 2 =45
OB = 45
F (6,3,2)
B (6,3,0)
45
© T Madas
x
y
z
A
DE
G
O
A cuboid OABCDEFG 6 units long by 3 units wide by 2 units high is drawn on a set of 3 dimensional axes as shown below.
1. Write down the coordinates of points B and F.
2. Calculate the length of OF.
6
3
2
C
O B
F
2
F (6,3,2)
B (6,3,0)
45
45
OF 2 =22+ 45 2
OF 2 = 4 + 45
OF 2 =49
OF = 7
© T Madas
© T Madas
7
24 32
The figure below shows a prism ABCDEF, whose cross sections are the right angled triangles ABC and DEF.
BAC = EDF = 90°, AB = 7 cm, AC = 24 cm, CF = 32 cm, M is the midpoint of BE and N is the midpoint of AD.
Calculate to an appropriate degree of accuracy:1. BC
2. AF
3. BFA
4. MFN
A
B
C
D
E
F
M
N
lengths in cm
© T Madas
7
24 32
The figure below shows a prism ABCDEF, whose cross sections are the right angled triangles ABC and DEF.
BAC = EDF = 90°, AB = 7 cm, AC = 24 cm, CF = 32 cm, M is the midpoint of BE and N is the midpoint of AD.
Calculate to an appropriate degree of accuracy:1. BC
2. AF
3. BFA
4. MFN
A
B
C
D
E
F
M
N
lengths in cm
A
B
C7
24
BC 2 =72+ 242
BC 2 =49+ 576
BC 2 =625
BC = 625BC = 25 cm
© T Madas
7
24 32
The figure below shows a prism ABCDEF, whose cross sections are the right angled triangles ABC and DEF.
BAC = EDF = 90°, AB = 7 cm, AC = 24 cm, CF = 32 cm, M is the midpoint of BE and N is the midpoint of AD.
Calculate to an appropriate degree of accuracy:1. BC
2. AF
3. BFA
4. MFN
A
B
C
D
E
F
M
N
lengths in cm
C
A
F
24
32
AF 2 =242 + 322
AF 2 =576+ 1024
AF 2 =1600
AF = 1600AF = 40 cm
© T Madas
7
24 32
The figure below shows a prism ABCDEF, whose cross sections are the right angled triangles ABC and DEF.
BAC = EDF = 90°, AB = 7 cm, AC = 24 cm, CF = 32 cm, M is the midpoint of BE and N is the midpoint of AD.
Calculate to an appropriate degree of accuracy:1. BC
2. AF
3. BFA
4. MFN
A
B
C
D
E
F
M
N
lengths in cm
A
B
F7
40θ
tanθ = 740
θ = 740
tan-1
θ ≈ 9.9°
© T Madas
7
24 32
The figure below shows a prism ABCDEF, whose cross sections are the right angled triangles ABC and DEF.
BAC = EDF = 90°, AB = 7 cm, AC = 24 cm, CF = 32 cm, M is the midpoint of BE and N is the midpoint of AD.
Calculate to an appropriate degree of accuracy:1. BC
2. AF
3. BFA
4. MFN
A
B
C
D
E
F
M
N
lengths in cm
7
N
F
24
16
NF 2 =242 + 162
NF 2 =576+ 256
NF 2 =832
NF = 832NF ≈28.844 cm
28.844
© T Madas
7
24 32
The figure below shows a prism ABCDEF, whose cross sections are the right angled triangles ABC and DEF.
BAC = EDF = 90°, AB = 7 cm, AC = 24 cm, CF = 32 cm, M is the midpoint of BE and N is the midpoint of AD.
Calculate to an appropriate degree of accuracy:1. BC
2. AF
3. BFA
4. MFN
A
B
C
D
E
F
M
N
lengths in cm
7
28.844
N
M
F7
28.844α
tanα = 728.44
α = 728.84
tan-1
α ≈ 13.6°
© T Madas
© T Madas
.56 57 cmA
B
C
D
E
F
20°40 cm40 cm
The diagram below shows a wedge in the shape of a right angled triangular prism.
FBC = 20°.R ABCD is a square of side 40 cmFind the angle the line AF makes with the plane
ABCD
x
x2 = 402 + 402 Ûx2 = 1600 + 1600 Ûx2 = 3200 Ûx = 3200
.» 56 57 cm
© T Madas
14.5
6 cm
.56 57 cmA
B
C
D
E
F
20°40 cm40 cm
The diagram below shows a wedge in the shape of a right angled triangular prism.
FBC = 20°.R ABCD is a square of side 40 cmFind the angle the line AF makes with the plane
ABCD
y40 = tan20° Û
y
y = 40 tan20° Û
y ≈ 14.56 cm
© T Madas
14.5656.57
14.5
6 cm
.56 57 cmA
B
C
D
E
F
20°40 cm40 cm
The diagram below shows a wedge in the shape of a right angled triangular prism.
FBC = 20°.R ABCD is a square of side 40 cmFind the angle the line AF makes with the plane
ABCD
= tanθ Û
θ
tanθ ≈ Û0.257
θ ≈ Ûtan-1 (0.257)
θ ≈ 14.4°
© T Madas
© T Madas
Experience on using the Pythagoras Theorem in 3 dimensions tells us that:
There are a few integer lengths which satisfy the Pythagorean law a 2 + b 2 + c 2 = d 2.
•Some of us are aware of 1,1,2,3•Even fewer of us know the 2,3,6,7•What about 3,4,12,13?•How many of us know the 4,5,20,21?•Does 51, 52, 2652,2653 also work?
What is the pattern of these quads?Is there an infinite number of 3D Pythagorean Quads?Is there a way to generate such numbers?
© T Madas
Prove that the product of any 2 consecutive positive integers a and b and their product c, satisfy the 3D Pythagorean relationship a 2 + b 2 + c 2 = d 2 , with d a positive integer
n 2+ (n + 1)2+ (n
2 + n)2 = n 2 + n
2+ 2n + 1+ n 4+ 2n
3+ n
Let a = n b = n + 1 c = n (n + 1) = n
2 + n
= n 4 + 2n
3+ 2n 2+ 3n + 1
= n 2 + n + 1( )2
© T Madas