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Chapter 2　Square roots
Square roots
Chapter 2
Chapter 2
●Key points of study●
1
Square roots　☞P 37〜
⑴　When the square of x is a, x is called the square root of a.
⑵　Positive numbers have two square roots with equal absolute value and different signs.
⑶　The square root of 0 is 0. Negative numbers do not have square roots.
2
How to represent square roots using radical signs　☞P 37〜
⑴　A positive square root is written as √ a (and read as “the square root of a”) and a negative one is written
as −√ a . When combined, they are written as ±√ a . The symbol √　is called a radical sign, or radix.
⑵　When a is a positive number,
2
①　（√ a ）
＝a
3
②　（−√ a ）2＝a
③　√ a2 ＝a
2
④　√（−a）
＝a
Relative size of square roots　☞P 38〜
⑴　If a and b are positive numbers and a＜b, then √ a ＜√ b .
⑵　The relationship between the size of √ a and b is equal to the relationship between the size of a and b2,
if b is a positive number.
4
Rational and irrational numbers, Recurring decimals　☞P 40〜
⑴　When a is an integer and b is a non-zero integer, any number that can be expressed as the fraction
a
is
b
called a rational number.
⑵　Any number that cannot be expressed as a fraction (of two integers) is called an irrational number.
The numbers √ 2 and ∏ are examples of irrational numbers.
⑶　Infinite decimals, in which a sequence of numbers is repeated after a certain number of decimal places,
are called recurring decimals. Such recurring decimals are rational numbers.
5
Multiplying and dividing expressions containing radical signs　☞P 42〜
When a and b are positive numbers,
①　√ a ×√ b ＝√ ab
6
②　√a
＝
√b
a
b
③　a√ b ＝√ a2b
④　√a
＝
b
a
b2
Rationalizing denominators　☞P 44〜
Eliminating radical signs in the denominator of a fraction is called rationalizing the denominator.
When b is a positive number,
7
a
a×√ b
a√ b
＝
＝
b
√b
√ b ×√ b
Adding and subtracting expressions containing radical signs　☞P 48〜
⑴　When more than one term in an expression contain the same square root, they can be simplified in the
same way as like terms.
①　m√ a ＋n√ a ＝
（m＋n）
√a
②　m√ a −n√ a ＝
（m−n）
√a
⑵　Calculate what is in √　or rationalize denominators to simplify expressions before adding or subtracting.
8
Approximate values of square roots worth remembering　☞P 39〜
hito - yo-hito-yo-ni-hito-mi-go-ro
⑴　√ 2 ＝1 . 4 1 4 2 1 3 5 6…
ni - yo-yo-ku-yo-ku
⑷　√ 6 ＝2 . 4 4 9 4 8 9…
hito - na-mi-ni -o-go-re-ya
⑵　√ 3 ＝1 . 7 3 2 0 5 0 8…
fu - ji-san-roku-o-mu-na-ku
⑶　√ 5 ＝2 . 2 3 6 0 6 7 9…
na - ni - mu-shi-i -na -i
⑸　√ 7 ＝2 . 6 4 5 7 5…
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Chapter 3　Quadratic equations
Let’s learn the basics 3 Problems about figures
Question In the figure on the right, rectangle EBFG has an area of 48 cm2. If its height is
twice that of square ABCD and its width is 2 cm shorter, then find the side length of square
ABCD.
Solution Letting the side length of the square be x cm, EB＝2x cm and BF＝x−2（cm）
.
Since the area of the rectangle is 48 cm2, 2x（x−2）＝48.
Solve this to get x＝−4，6.
Considering x＞2, the answer is x＝6.
Answer 6 cm
9
A class had a square flowerbed that was x m on one side. When a new
□
rectangular flowerbed was formed by making the original flowerbed 2 m shorter
and 3 m wider, the resulting area became 24 m2. Find the value for x.
10
There is a metal plate 24 cm wide as shown on the right. If its left and right
□
sides are folded up by the same amount to form a rain gutter with a crosssectional area of 54 cm2, how many cm should be folded up on each side?
11 As shown on the right, a fixed-width flowerbed is placed around (3 sides) of a
□
rectangular plot of land that measures 20 m by 30 m. The remaining part of the land
is covered by a lawn that takes up 68％ of the total area. Find the width of the
flowerbed
lawn
flowerbed.
12
As shown on the right, four squares, each with a side length of 3 cm, were cut
□
from the corners of a square sheet of paper to form a cuboid-shaped container whose
volume was 675 cm3. Find the side length of the original square paper.
13
The side length of square ABCD on the right is 15 cm. Point P moves from A to
□
B along side AB at a speed of 1 cm per second. Starting at the same time and
traveling at the same speed, point Q moves from B to C along side BC. How many
seconds after P and Q depart will the area of ▲PBQ become 28 cm2?
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Chapter 7　The Pythagorean theorem
The Pythagorean theorem
Chapter 7
●Key points of study●
1
The Pythagorean theorem and the converse of the Pythagorean theorem　☞P 167〜
⑴　When the lengths of two sides adjacent to the right angle in a right triangle are a and b
and the length of the hypotenuse is c, the relationship between the lengths of the three
sides is a2+b2=c2. (the Pythagorean theorem)
⑵　When the lengths of the three sides of a triangle are a, b, and c and the relationship
between them is a2+b2=c2, the triangle is a right triangle in which the length of the
hypotenuse is c. (the converse of the Pythagorean theorem)
2
Ratio of three sides in a special triangle　☞P 171〜
3
The Pythagorean theorem and plane figures　☞P 171〜
Chapter 7
⑴　Length of diagonals in a square
Height and area of an
⑵　Length of diagonals in a rectangle ⑶　x= 2 a
x= a2+b2
equilateral triangle
3
3 2
h=
a，S=
a
2
4
⑷　Distance between two points
⑸　Length of a chord in a circle
⑹　Length of a tangent to a circle
AB= （x1−x2）+
（y1−y2）
¬=2 r −d
¬= d 2−r 2
2
4
2
2
2
The Pythagorean theorem and solids　☞P 180〜
⑴　Length of diagonals in a cuboid ⑵　Height of a square pyramid
⑶　Height of a cone
x= a2+b2+c2
OH= OA2−AH2
h= ¬ 2−r 2
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Chapter 7　The Pythagorean theorem
Let’s learn the basics 2 Proving that the Pythagorean theorem is true
Question There are four congruent right triangles in square ABCD on the right. The
lengths of the two sides adjacent to the right angle in each triangle are a and b and
the length of each respective hypotenuse is c. Using this figure, prove that the
Pythagorean theorem is true.
Answer The total area of the four triangles is
1
ab×4=2ab.
2
Quadrilateral EFGH is a square whose side length is a−b,
so its area is（a−b）2.
The total area of the four triangles and square EFGH is
2
2ab+
（a−b）
=2ab+
（a2−2ab+b2）
=a2+b2. ……①
The side length of square ABCD is c, so its area is c2. ……②
Since ① and ② are equal, it is true that a2+b2=c2.
3
There are two congruent right triangles in trapezoid ABCD on the right. The lengths
□
of the two sides adjacent to the right angle in each triangle are a and b and the length
of each respective hypotenuse is c. When using this figure to prove that the
Pythagorean theorem is true, fill in the blanks below to complete the proof.
〔Proof〕　The area of trapezoid ABCD is expressed in terms of a and b as ⅰ .
The total area of
ⅱ .
ABE is expressed in terms of c as ⅲ .
1
Since ⅳ − ⅴ = c2,
2
2
2
it is true that a +b =c2.
ADE and
ECB is expressed in terms of a and b as
The area of
4
In the figure on the right, ABC is a right triangle in which ∠ACB=90°, and
□
each its side is shared by one of the following squares: ADEB, BFGC, ACIJ.
The straight line drawn from C intersects AB perpendicularly at point H and
intersects DE at point K. Fill in the blanks below to complete the proof that
AC2+BC2=AB2.
Since CJ is a diagonal of a square, square ACIJ=2
ACJ. …①
Since they share the same base and have the same height,
ACJ= ⅰ . …②
Since two pairs of sides and the angle between them are equal,
Since they share the same base and have the same height,
ABJ≡ ⅱ .…③
ADC= ⅲ . …④
Since DH is a diagonal of a rectangle, rectangle ADKH=2 ⅳ . …⑤
It can be said from ① to ⑤ that square ACIJ=rectangle ⅴ . …⑥
Similarly, square BFGC=rectangle ⅵ . …⑦
It can be said from ⑥ and ⑦ that square ACIJ+square BFGC=square ⅶ .
Therefore, it is true that AC2+BC2=AB2.
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