Angles and Parallel Lines (II) 6.13
Diagram Clue
A
B
C
F E
D
T
x
y
Q
M
P
L
N
R
a
cb
➥ Ex 6A 13-16
Rates, Ratios and Proportions 8.37
direct proportion 正比例
The table below shows some corresponding values of the number of
oranges sold (x) and the total income (y) of a hawker.
x 1 5 10 20 30
y $6 $30 $60 $120 $180
1. From the table above,
the greater the number of oranges sold, the ( more / less ) the total income.
2. (a) For each pair of values of x connected with an arrow below, complete the ratio.
x 1 5 10 20 30
(b) For each corresponding pair of values of y, complete the ratio.
y $6 $30 $60 $120 $180
3. From Question 2,
the ratio of each pair of values of x ( = / ! ) the ratio of the corresponding pair of values of y.
ActivityActivity
Class 8.2
1 : 5 1 :
:
1 :
1 :
:
C Direct Proportions
From Class Activity 8.2, we can see that as the number of oranges
sold (x) increases, the total income (y) of the hawker also increases; and
more importantly,
1 : 5 = $6 : $30
1 : 10 = $6 : $60
20 : 30 = $120 : $180
We note that the ratio of the numbers of oranges sold (x) is equal to the
ratio of the corresponding total incomes (y). We say that x and y are in
direct proportion, or x is directly proportional to y.
x 1 5 10 20 30
y $6 $30 $60 $120 $180
1 : 5
1 : 10
2 : 3
1 : 5
1 : 10
2 : 3
8.38 Chapter 8
In general:
In direct proportion, the ratio of two values of a quantity is equal to
the ratio of two corresponding values of another quantity.
Suppose the quantities x and y are in direct proportion, where x1, y1 and
x2, y2 are two pairs of values of x and y.
Then,
x1 : x2 = y1 : y2
` 2
1
x
x = 2
1
y
y
i.e.
1
1
y
x = 2
2
y
x
Note: In fact, when x and y are in direct proportion, y
x is a constant.
x x1 x2
y y1 y2
x and y are in direct proportion. When x = 8, y = 12.
When x = 10, find the value of y.
12
8 = y
10
8y = 12 # 10
y = 15
Level 1
Example 20
Solution x 8 10
y 12 ?
◀
x and y are in direct proportion. When x = 15, y = 9.
(a) When y = 6, find the value of x.
(b) When x = 20, find the value of y.
20
Instant Drill
➥ Ex 8D 1-5
Let's illustrate more applications of direct proportion in Example 21 and
Example 22. If 3 pens are sold for $18, find the price of 5 pens.
Level 1
Example 21
續三角形 7.7
(b) a 3DBE , 3DCF
` +DBE = +DCF
` AB = AC
即 3ABC 是一個等腰三角形。
在 (a) 部已證
全等 3 的對應角
等角對邊相等
在圖中,D 是 AB 上的一點。
(a) 證明 3ABC + 3CBD。
(b) 若 +CBD = +CDB,證明 3ABC
是一個等腰三角形。
4
即時訓練即時訓練
➥ 習題 7A 26, 27
B D
30 cm
20 cm
25 cm
C
A
在圖中,S 是 PR 上的一點,使 +PRQ = +PQS。
(a) 證明 3PQR + 3PSQ。
(b) 若 PS = 9 cm,PQ = 15 cm 和 QR = 20 cm,證明
3PSQ 是一個直角三角形。
(a) 在 3PQR 和 3PSQ 中,
+PRQ = +PQS
+QPR = +SPQ
+PQR = 180c - +PRQ - +QPR
+PSQ = 180c - +PQS - +SPQ
= 180c - +PRQ - +QPR
` +PQR = +PSQ
` 3PQR + 3PSQ
已知
公共角
3 內角和
3 內角和
AAA
程度二 5例例
Q
P
R
S
解
AAA
Q Q
S
P
R
P
公共角
1.48 第 1 章
整除性與閏年
我們常說一年有365 天。然而,因為地球需要大約365.242 2 天
圍繞太陽一周,所以幾乎每四年便會多出一天(即 2 月 29 日 )。
有 366 天的年份稱為閏年。
閏年必定符合以下其中一個規則:
1. 該年份可以被 4 整除,但不可以被 100 整除。
2. 該年份可以被 400 整除。
例如,2020 年、2024 年和 2400 年都是閏年,而 2021 年、2100 年
和 2200 年則不是閏年。
利用整除性判別方法,判斷下列各年份是否閏年。
(a) 2036 (b) 2158
(c) 2322 (d) 2476
考考你
數學的應用
ET MS
全等及相似(二)9.9
都是直線。若
和 +ACB = 30c,證明
E
30°
D C
,
30c
30c
已知
公共角
3 內角和
3 內角和
AAA
➥ 習題 9A 8, 9
7
8
6
A
B
C
4
3.5
E
F
D
x
B。
B
60°
A
Q
P
C
C
A
B
D C
E
公共角
AAA
8.42 Chapter 8
In the figure, ABCD is a square. AOCE and BOD are straight
lines.
(a) Prove that BE = DE.
(b) If DE = DB, find +EBC.
(a) In 3BOE and 3DOE,
BO = DO
a AC = BD
` +BOE = +DOE = 90c
OE = OE
` 3BOE , 3DOE
` BE = DE
property of square
property of square
common side
SAS
corr. sides, , 3s
(b) DE = DB (given)
BE = DE (proved in (a))
` 3DBE is an equilateral triangle.
` +DBE = 60c (property of equil. 3)
+DBC = 45c (property of square)
` +EBC = +DBE - +DBC
= 60c - 45c
= 15c
Level 2
Example 17
D
A B
E
O
C
Solution
In the figure, ABCD is a rectangle and
AFEC is a straight line. It is given that
17
Instant Drill
A B
F
角和平行線 (二)
2A冊 第6章
5
四邊形
3B冊 第8章
續三角形
3A冊 第7章
以生活情境引入正比例的概念
Book 2B Ch.8•P.8.37-8.44
探究整除性的相關應用 第1.48 頁
中三
• 涉及角、平行線和三角形的較複雜的證明題
• 涉及四線、四心和四邊形的綜合題
小販收入
與正比例
延伸應用