Download A. Pythagoras` Theorem

11
Pythagoras’ Theorem
Case Study
11.1 Pythagoras’ Theorem and Its Proofs
11.2 Applications of Pythagoras’ Theorem
11.3 Converse of Pythagoras’ Theorem and
Its Applications
11.4 Surds and Irrational Numbers
Chapter Summary
Case Study
We have to know the height
of the slide and the horizontal
distance between the top and
the bottom of the slide.
How can we find the
length of the slide?
In the figure, y is the length of the slide, h is the height of the slide and
x is the horizontal distance between the top and the bottom of the slide.
By using Pythagoras’ theorem, we know that y2  h2  x2.
As a result, we can find the length of the slide once we know the values
of x and h.
P. 2
11.1 Pythagoras’ Theorem and Its Proofs
A. Pythagoras’ Theorem
The figure shows a right-angled triangle ABC, where C  90.
AB is the longest side of the right-angled triangle.
It is called the hypotenuse (the side opposite to the
right angle) of the triangle.
Moreover, the side opposite to an angle is usually named by its
corresponding small letter:



a : opposite side BC of A
b : opposite side AC of B
c : opposite side AB of C
P. 3
11.1 Pythagoras’ Theorem and Its Proofs
A. Pythagoras’ Theorem
An important relationship among the 3 sides of a right-angled triangle:
In a right-angled triangle, the sum of the squares
of the 2 shorter sides is equal to the square of
the hypotenuse. That is, in DABC, if C  90,
then a2  b2  c2.
(Reference: Pyth. theorem)
The above result is known as the Pythagoras’ theorem.
We are going to discuss some methods in proving the Pythagoras’ theorem
in the next section.
P. 4
11.1 Pythagoras’ Theorem and Its Proofs
A. Pythagoras’ Theorem
Example 11.1T
In DXYZ, Y  90, XY  16 and XZ  34.
Find the value of a.
Solution:
In DXYZ,
XY 2  YZ 2  XZ 2
162  a2  342
a2  1156  256
 900
∴
a  900
 30
P. 5
(Pyth. theorem)
11.1 Pythagoras’ Theorem and Its Proofs
A. Pythagoras’ Theorem
Example 11.2T
In DXYZ, X  90, XY  2 and YZ  3. Find the
value of f. (Give the answer in surd form.)
Solution:
In DXYZ,
XY 2  XZ 2  YZ 2
22  f 2  32
f2  9  4
5
∴
f 5
P. 6
(Pyth. theorem)
11.1 Pythagoras’ Theorem and Its Proofs
A. Pythagoras’ Theorem
Example 11.3T
In the figure, DXYZ is a right-angled triangle with Y  90, XY  21 cm,
WZ  8 cm and XZ  11 cm.
(a) Find YW and XW.
(b) Find the perimeter of DXWZ.
Solution:
(a) In DXYZ,
XY 2  YZ 2  XZ 2
∴
YZ  112  ( 21) 2 cm
 100 cm
 10 cm
∴
YW  (10  8) cm
 2 cm
P. 7
(Pyth. theorem)
11.1 Pythagoras’ Theorem and Its Proofs
A. Pythagoras’ Theorem
Example 11.3T
In the figure, DXYZ is a right-angled triangle with Y  90, XY  21 cm,
WZ  8 cm and XZ  11 cm.
(a) Find YW and XW.
(b) Find the perimeter of DXWZ.
Solution:
(a) In DWXY,
YW 2  XY 2  XW 2
(Pyth. theorem)
∴
XW  22  ( 21) 2 cm
 25 cm
 5 cm
(b) Perimeter of DXWZ  XW  WZ  XZ
 (5  8  11) cm
 24 cm
P. 8
11.1 Pythagoras’ Theorem and Its Proofs
B. Different Proofs of Pythagoras’ Theorem
There is evidence that the ancient Chinese and Indians applied the
concept of the Pythagoras’ theorem in early time.
However, Pythagoras was the first to discover the theorem by geometric
proof.
Some methods in proving the Pythagoras’ theorem:
 Proved in Ancient Greece
 By Zhao Shuang in Ancient China
 By James Garfield in the United States
P. 9
Pythagoras probably
proved the theorem in
another way. Please
refer to Enrichment
Mathematics, p.205 of
Book 2B for details.
11.1 Pythagoras’ Theorem and Its Proofs
B. Different Proofs of Pythagoras’ Theorem
(a) Proved in Ancient Greece
Each figure has 4 identical right-angled triangles (in purple colour)
∵ The areas of the 2 figures are equal.
∴ a2  b2  c2
P. 10
11.1 Pythagoras’ Theorem and Its Proofs
B. Different Proofs of Pythagoras’ Theorem
(b) By Zhao Shuang in Ancient China
In ancient China, mathematicians proposed the
Gougu theorem（勾股定理）to express the
relationship among the 3 sides of a right-angled
triangle, where the base and the height of the
triangle were named as Gou（勾）and Gu（股）
respectively.
In about 350 AD, Zhao Shuang used the Yuan-Dao（弦圖）to
prove the theorem in the Chou Pei Suan Ching《周脾算經》.
P. 11
11.1 Pythagoras’ Theorem and Its Proofs
B. Different Proofs of Pythagoras’ Theorem
(b) By Zhao Shuang in Ancient China
A Yuan-Dao consists of 4 identical right-angled triangles with
sides a, b, c and a square with side (b  a).
4
∵
∴

Area of ABCD  4  Area of DADH  Area of EFGH
1
c2  4  ab  (b  a)2
2
 2ab  b2  2ab  a2
 a2  b2
P. 12
11.1 Pythagoras’ Theorem and Its Proofs
B. Different Proofs of Pythagoras’ Theorem
(c) By James Garfield in the United States
The 20th president of the United States, James Garfield (1831  1881)
developed his own proof in The Journal of Education (Volume 3
issue 161) in 1876:
In the figure, trapezium ABCD is formed by 2 congruent
right-angled triangles ABE and ECD, and an isosceles
right-angled triangle AED.
∵
∴
Area of trapezium  2  Area of DABE  Area of DAED
1
1
1
(a  b)2  2  ab  c2
2
2
2
(a  b)2  2ab  c2
a2  b2  c2
P. 13
11.2 Applications of Pythagoras’ Theorem
In our daily lives, we often come across problems that can be
modelled as right-angled triangles.
Pythagoras’ theorem can be used to solve problems involving
right-angled triangles.
P. 14
11.2 Applications of Pythagoras’ Theorem
Example 11.4T
A ladder of length 3.2 m leans against a vertical wall. The distance of
its top from the ground is the same as the distance of its foot from the
wall. Find the distance of the bottom of the ladder from the wall.
(Give the answer correct to 1 decimal place.)
Solution:
Let x m be the distance of bottom of the ladder from
the wall.
∴
∴
x2  x2
2x2
x2
x
 3.22
(Pyth. theorem)
 10.24
 5.12
 5.12
 2.3 (cor. to 1 d. p.)
The distance of the bottom of the ladder from the wall is 2.3 m.
P. 15
11.2 Applications of Pythagoras’ Theorem
Example 11.5T
The figure shows a trapezium with AD // BC, A  B  90.
If AB  12 cm, AD  15 cm and BC  20 cm, find the
perimeter of the trapezium.
E
Solution:
As shown in the figure, construct a line DE such that DE  BC.
∴ DE  12 cm and EC  5 cm
In DCDE,
CE 2  DE 2  CD 2
(Pyth. theorem)
∴
CD  52  122 cm
 13 cm
∴ Perimeter of the trapezium  AB  BC  CD  DA
 (12  20  13  15) cm
 60 cm
P. 16
11.2 Applications of Pythagoras’ Theorem
Example 11.6T
Peter and Lily left school at 4:00 p.m. Peter walked east at a speed of
2.4 m/s to reach the library at 4:15 p.m. Lily walked north at a speed
of 2.25 m/s to reach the bookstore at 4:12 p.m.
(a) How far did each of them walk?
(b) Find the distance between the library and the bookstore.
Solution:
(a)
Distance travelled by Peter
 (2.4  15  60) m
 2160 m
(b) As shown in the figure,
Distance travelled by Lily
 (2.25  12  60) m
 1620 m
AB 2  AC 2  BC 2
(Pyth. theorem)
∴
BC  21602  16202 m
 2700 m
∴ The distance between the library and the bookstore is 2700 m.
P. 17
11.3 Converse of Pythagoras’ Theorem
and Its Applications
In the previous section, we learnt the Pythagoras’ theorem.
In fact, the converse of Pythagoras’ theorem is also true and is
stated below:
In a triangle, if the sum of the squares of the 2
shorter sides is equal to the square of the longest
side, then the triangle is a right-angled triangle
with the right angle opposite to the longest side.
That is, in DABC, if a2  b2  c2,
then C  90.
(Reference: converse of Pyth. theorem)
P. 18
11.3 Converse of Pythagoras’ Theorem
and Its Applications
Example 11.7T
In the figure, XZ  25, YZ  30 and XW  20. W is the mid-point of YZ.
(a) Prove that XWZ  90.
(b) Prove that DXYZ is an isosceles triangle.
Solution:
(a) In DXWZ,
(b) In DXWY,
WZ  30  2
WY  15
 15
XWY  90
WX 2  WZ 2  152  202
WX 2  WY 2  XY 2 (Pyth. theorem)
 225  400
XY  152  202
 625
 25
2
2
XZ  25
 XZ
 625
∴ DXYZ is an isosceles triangle.
2
2
2
∵ WX  WZ  XZ
∴ XWZ  90 (converse of Pyth. theorem)
P. 19
11.3 Converse of Pythagoras’ Theorem
and Its Applications
Example 11.8T
The figure shows a piece of triangular paper. It is known that XW  YZ,
XW  12 cm, YW  9 cm and WZ = 16 cm.
(a) Find XY and XZ.
(b) Prove that the paper is in the shape of
a right-angled triangle.
Solution:
(a) In DWXY,
In DWXZ,
WX 2  WY 2  XY 2 (Pyth. theorem) WX 2  WZ 2  XZ 2 (Pyth. theorem)
XY  122  92 cm
XZ  122  162 cm
 15 cm
 20 cm
(b) In DXYZ,
XY 2  XZ 2  152  202
∵ XY 2  XZ 2  YZ 2
 625
∴ YXZ  90 (converse of Pyth. theorem)
YZ 2  (9  16)2
∴ The paper is in the shape of a
 625
right-angled triangle.
P. 20
11.4 Surds and Irrational Numbers
A. Surds on a Number Line
In Book 1A Chapter 1, we learnt how to represent real numbers on
a number line.
We can also represent surds on a number line.
For example, to represent 2 on a number line, first construct a
right-angled triangle OAB with OA  1 unit and AB  1 unit.
∴
OB  12  12 units
 2 units
C
2
Using a pair of compasses,
draw an arc with centre O and radius OB to meet the number line at C.
∵
∴
OC  OB
OC  2 units
P. 21
(radii)
11.4 Surds and Irrational Numbers
A. Surds on a Number Line
Example 11.9T
Represent 13 on a number line.
Solution:
Step 1: Consider the number 13 as the sum of 2 perfect squares,
i.e., 32  22  13.
Step 2: Construct a right-angled triangle with
OA  3 units and AB  2 units.
Step 3: Draw an arc with centre O and
radius OB to meet the number
line at C. We have OC  13 .
P. 22
C
13
11.4 Surds and Irrational Numbers
A. Surds on a Number Line
Example 11.10T
Represent 11 on a number line.
Solution:
Step 1: Consider the number 11 as the sum of perfect squares,
i.e., 32  12  12  11.
Step 2: Construct a right-angled triangle
with legs 3 units and 1 unit.
Then mark 10 ( 32  12 ) on
the number line.
Step 3: Continue to construct a
right-angled triangle with legs
10 units and 1 unit. Then
mark 11 on the number line.
P. 23
10
11.4 Surds and Irrational Numbers
B. First Crisis of Mathematics
At the time of Pythagoras, people believed that all things could be
explained by numbers which are either integers or fractions.
The discovery of 2 shocked the society and lead to the first crisis of
mathematics.
After the Pythagoras’ theorem was proved, a follower
of Pythagoras, Hippasus of Metapontum (about 500 BC)
tried to demonstrate the length of the diagonal of a
square with side 1 unit by applying the theorem.
However, he found that such length (the square root of 2) was neither
an integer nor a fraction.
His discovery went against the belief of the Greeks, especially the
Pythagoreans.
They did not accept the existence of irrational numbers.
P. 24
11.4 Surds and Irrational Numbers
B. First Crisis of Mathematics
Legend reveals that Hippasus was then caught and sentenced to death
by drowning.
Although Hippasus was executed, Pythagoras admitted the existence
of irrational numbers.
It was only until 2000 years later that the irrational numbers were
defined using the concept of rational numbers.
P. 25
Chapter Summary
11.1 Pythagoras’ Theorem
In a right-angled triangle ABC,
a2  b2  c2
(Reference: Pyth. theorem)
There are many different proofs of the Pythagoras’ theorem.
1. Proved in Ancient Greece
2. By Zhao Shuang in Ancient China
3. By James Garfield in the United States
P. 26
Chapter Summary
11.2 Applications of Pythagoras’ Theorem
In our daily lives, we often come across problems that can be modelled
as right-angled triangles.
Such problems can be solved by the Pythagoras’ theorem.
P. 27
Chapter Summary
11.3 Converse of Pythagoras’ Theorem and Its
Applications
In DABC,
if a2  b2  c2,
then C  90.
(Reference: converse of Pyth. theorem)
P. 28
Chapter Summary
11.4 Surds and Irrational Numbers
1.
We can represent surds on a number line.
2.
The discovery of
P. 29
2 caused the first crisis of mathematics.