Page 1 of 26

Secondary 2 Textbook Answer Key

Chapter 1 Real Numbers and Its Operations

1.1 Square Roots

A. Definition of Square Roots

(Page 3)

（1）±11 （2）0 （3）±5

2

（4）No. Because the square of any real number is non-negative.

B. Definition of Principal Square Root

(Page 5)

1. （1）±4 （2）±0.5 （3）±9

5

2. (1) False. There are two solutions: ±0.3.

(2) False. √25 = 5.

C. Application of Principal Square Root

(Page 6)

1. 2 2. 15𝑐𝑚

1.2 Cube Roots

A. Definition of Cube Roots

1. Definition

(Page 7)

（1）3 （2）−6 （3）0

2. Find the cube root of a real number

(Page 8)

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（1）8 （2）−0.3 （3）−5

4

B. Application to Cube Roots

(Page 8)

7𝑐𝑚

1.3 Understanding Real Numbers Again

A. Understanding 4th Root and nth Root

(Page 9)

(1) 2, −2 (2) 1, 2 (3) 0, 0

(Page 10)

(1) 6 (2) 3 (3) −3 (4) 1

2

B. Real Numbers and Number Line

(Page 12)

1.

2. On a number line, 5 is between 2 and 3.

1.4 Operations on Quadratic Radicals

1.4.1 Definition of Quadratic Radicals

(Page 13)

1. （1）8 （2）9 （3）9 （4）8

2.（1）When 𝑥 ≥ −1. （2）When 𝑥 ≥2

3.

3. No. (√𝑎)2

= 𝑎, while √𝑎2 = |𝑎|

1.4.2 Multiplying and Dividing Quadratic Radicals

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A. Multiplying Quadratic Radicals

(Page 15)

(1) 5√2 (2) 3√5𝑎

B. Dividing Quadratic Radicals

(Page 17)

1. （1）4√3 （2）5𝑎√𝑎 （3）√5

5 （4）

√6

3

2. （1）7√6 （2）3𝑏√2 （3）√10

5

（4）√15

5

3. 5√2

2

C. Simplifying Quadratic Radicals

(Page 18)

1．（1） 64 （2）222 x （3） 72

（4）

3

6

2．（1） 156 （2）234 ab （3）

2

1 （4）

2

x

1.4.3 Adding and Subtracting Quadratic Radicals

(Page 20)

1．（1）Yes. （2）No. （3）Yes. （4）No.

2．√50, √1

2．

3． （1）7√3

4. （2）−10√3．

1.4.4 Mixed Operations on Quadratic Radicals

(Page 22)

（1）−1 （2）24√3 − 6√10 （3）12√3 − 60

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Chapter 2 Polynomial Operations

2.1 Adding and Subtracting Polynomials

A. Definition of Polynomials

(Page 25~26)

1. （1）False （2）False （3）True

2.（1）Degree: 2. Terms: 3. （2）Degree: 2. Terms: 3.

3.（1）Degree: 2. Terms: 2. （2）Degree: 3. Terms: 4.

B. Like Terms

Page (27)

1. 322 cab . Infinitely many. Yes.

2. When 𝑘 = 2,kyx 323 and

622 yx are like terms.

3.

C. Combining Like Terms

(Page 29)

1．0

2．（1）Liker terms：3𝑥 and −𝑥, −𝑥² and 3𝑥², 5 and −5.

The result after combing the like terms: 2𝑥2 + 2𝑥.

（2）Liker terms: 3𝑎² and −3𝑎², −5𝑏² and 4𝑏².

The result after combing the like terms: −𝑏² + 2𝑎𝑏.

3．（1） 322 2 xx . When 𝑥 = −2, the result is 7.

（2） 12 ba . When 2,1 ba , the result is −5.

Page 5 of 26

D. Removing and Inserting Parenthesis

(Page 31)

1.（1）𝑎 − 𝑏 − 𝑐 − 𝑑 (2) 𝑎 − 𝑏 + 𝑐 − 𝑑

(3) −𝑎 + 𝑏 + 𝑐 − 𝑑 (4) −𝑎 + 𝑏 + 𝑐 − 𝑑

2.（1）22 2 baba （2）

22 75 yx （3） 2215 abba

(Page 33)

1.（1）= 116𝑥 + (138𝑥 − 38𝑥) = 116𝑥 + 100𝑥 = 216𝑥

（2）= 128𝑥 − (64𝑥 + 36𝑥) = 128𝑥 − 100𝑥 = 28𝑥

（3）= 135𝑥 − (87𝑥 − 57𝑥) = 135𝑥 − 30𝑥 = 105𝑥

2. (1) 𝑥𝑦2 − 2𝑦2 (2) (−𝑎3 + 2𝑎2) − (3𝑎 − 1)

(3) 3𝑥2𝑦2 − (𝑥3 − 2𝑦3)

E. Adding and Subtracting Polynomials

(Page 35)

1.（1）32 yx （2） 93 x

2．（1）22 32 ba . When 𝑎 =

3

1, 𝑏 = 3, the result is −27

2

9.

（2）22 59 xyyx . When 𝑥 =

1

2, 𝑦 = −1, the result is −

19

4.

2.2 Multiplying Polynomials

A. Multiplying Monomials

(Page 36)

1.（1） −2𝑎5 （2） −72𝑎3𝑏5 （3）−108𝑎12 （4）−

1

2𝑥5𝑦4𝑧

2. 1.5 × 1011𝑚

B. Multiplying with Monomial and Polynomial—Multiplication

Involving 𝒂(𝒎 + 𝒏 + 𝒑)

(Page 37)

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1. （1）12𝑥 4 𝑦 3 − 6𝑥 4 𝑦 2 （2）−6𝑥3 + 9𝑥2𝑦 + 6𝑥𝑦2

2. −6𝑥3 + 4𝑥 + 2𝑥3 − 4𝑥4 − 6𝑥2 + 10𝑥

C. Multiplication with Polynomial and Polynomial—Multiplication

Involving (𝒂 + 𝒃)(𝒎 + 𝒏)

(Page 39)

1. （1）6𝑥2 + 𝑥 − 12 （2）𝑥2 + 2𝑥𝑦 − 15𝑦2

（3）4𝑚2 − 9𝑛2 （4）4𝑎2 + 12𝑎𝑏 + 9𝑏2

2. 2(𝑎 + 2𝑚)(𝑏 + 𝑚) + 𝑎𝑐 = 2𝑎𝑏 + 2𝑎𝑚 + 4𝑏𝑚 + 4𝑚² + 𝑎𝑐

2.3 Multiplication Formulas

A. Square of Sums or Difference of Two Numbers—Multiplication

Involving (𝒂 ± 𝒃)𝟐

(Page 41)

1. （1）4𝑥 2 −1

9 （2）9𝑥2 − 4

（3）25𝑦2 − 4𝑥2 （4）4𝑥2 − 𝑦2

2. (1) = (500 − 2)(500 + 2) = 5002 − 4 = 249996

(2) = (2000 − 5)(2000 + 5) = 20002 − 25 = 3999975

B. Multiplication with sum and Difference of two Numbers—

Multiplication Involving (𝒂 + 𝒃)(𝒂 − 𝒃)

(Page 43)

1. （1）9𝑥2 + 6𝑥 + 4. （2）𝑛2 − 4𝑚𝑛 + 4𝑚2

2. （1）𝑥2 − 4𝑥 + 4. （2）4𝑚2 + 12𝑚𝑛 + 9𝑛2

3. (𝑎 + 0.1)2 = 𝑎2 + 0.2𝑎 + 0.01 meters.

2.4 Dividing Polynomials

2.4.1 Dividing Monomials

(Page 44)

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1.

Dividend 6𝑥 3 𝑦 3 −42𝑥 3 𝑦 3 −42𝑥 3 𝑦 3

Divisor 2𝑥𝑦 −6𝑦 3 −6𝑥 2 𝑦 2

Quotient 3𝑥 2𝑦

2 7𝑥 3 7𝑥𝑦

2. (3 × 10 8 ) ÷ (3.4 × 10 2 ）≈ 882300

2.4.2 Dividing a Polynomial by a Monomial

(Page 45)

1.（1）3𝑏 − 4 （2）−3𝑎𝑥 − 15 （3）3𝑚 −4

15𝑛 （4）−

2

1𝑥 + 𝑦

2.（1）−2𝑎2𝑏 + 3𝑎𝑏2𝑐 + 1 （2）−𝑥𝑦 + 2𝑥2 − 4𝑥

2.4.3 Dividing a Polynomial by a Polynomial

(Page 47)

(1)

1-2

121 2

x

xxx

xx 22 2

1-- x

1-- x

0

∴ )1()1-2( 2 xxx = 12 x

（2） 343x2 3 xx

xx

xxx

62

4-323-

2

3

23 6-2 xx

436 2 xx

xx 18-6 2

4-21x

∴ 343x2 3 xx = xx 62 2 ..........R ( 4-21x )

Page 8 of 26

2.5 Factoring

2.5.1 Factoring by Common Factor—Factoring Involving 𝒂𝒎 + 𝒂𝒏 +

𝒂𝒑

(Page 49)

1. （1）False.(2𝑎 − 1)2 (2) True.

2. （1） 𝑎(𝑎 + 1) （2）2𝑎𝑏(2 − 𝑎)

（3） 2𝑎(𝑚2 − 4) （4） ）（ nmp 56)3(

2.5.2 Difference of Squares—Factoring Involving 𝒂𝟐 − 𝒃𝟐

(Page 50)

)42(8.5

)3()3(.4)4.0()4.0(.3

)11

2

3

1)(

11

2

3

1(.25)5.1

nmm

cbacbaabab

nmnmxx

）（（

2.5.3 Perfect Square Trinomials—Factoring Involving 𝒂𝟐 ± 𝟐𝒂𝒃 + 𝒃𝟐

(Page 52)

（1）−(𝑎 − 3)2 （2）−(4𝑎 + 𝑏)2 （3）−𝑎(𝑎 − 1)2

2.5.4 Introduction to Factoring Higher Order Polynomials

(Page 54)

(1) 𝑎𝑏(𝑎² + 1) (2) 5𝑥²𝑦(𝑦² − 2)

(3) 3𝑎(𝑥 + 𝑦)2 (4) (𝑥2 + 𝑦2 )(𝑥 + 𝑦)(𝑥 − 𝑦)

2.6 Applications to Polynomial Operations

(Page 57)

1. It takes 4 + 3(𝑛 − 1) = 3𝑛 + 1 matches to make 𝑛 square.

When 𝑛 = 2012, it takes 3 × 2012 + 1 = 6037 matches.

2. In the column, there is a difference of 16 between two numbers next to

Page 9 of 26

each other. In the row, there is a difference of 2 between two numbers next

to each other.

3. 𝑛

2𝑛+1.

4．A. (1) 1 triangle (2) 4 triangles (3) 7 triangles

B. 3𝑛 − 2

Page 10 of 26

Chapter 3 Geometry

3.1 Plane and Solid

A. Understanding the Definition of Plane and Solid by Real Life

Examples

(Page 61)

Examples for plane shapes: computer screen, blackboard, desk top

Examples for solid shapes: book, wood block, basketball

B. Expansion of Solid Shapes

(Page 64)

5, 2. 4.

3.2 Surface Area of a Solid

A. Surface Area of Cylinder and Cone

(Page 65)

1. 8𝜋

2. 8𝜋

B. Surface Area of Cuboid

(Page 65)

1. 3 2. 4

C. Surface Area of Sphere

(Page 66)

1. 100𝜋 2. 4

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3.3 Volume of Solid

A. Volume of Cylinder and Cone

(Page 68)

1.（1）72𝜋 （2）3

2.（1） π3

16 （2）2

B. Volume of Prism

(Page 69)

1. 40 2. 4 3. 27

C. Volume of Sphere and Pyramid

(Page 70)

1. （1） 3

500V （2）1

2. （1）240 （2）384𝑐𝑚³

D. Operations Involving Different Solids of Same Volume

(Page 71)

1. 6. 2. 3

34 3. 15

3.4 Congruence of Triangles

3.4.1 Recognizing Congruence Triangles and Their Properties

(Page 73)

1. 𝐴𝐵 = 𝐶𝐷, 𝑂𝐵 = 𝑂𝐷, 𝑂𝐴 = 𝑂𝐶, ∠𝐴 = ∠𝐶, ∠𝐵 = ∠𝐷, ∠𝐴𝑂𝐵 = ∠𝐶𝑂𝐷

2. 70° 3. 11

Page 12 of 26

3.4.2 Criteria of Congruence Triangles

A. Side-Side-Side (SSS) Congruence and Compass and Straightedge

Construction on an Angle That Equals to a Given Angle

(Page 77)

1. ∠𝐵 = ∠𝐴 = 92°. (Hint: Connect CD and prove that △ 𝐶𝐴𝐷 ≌△ 𝐶𝐵𝐷

（SSS） to obtain.)

2．Solution. Since 𝐵𝐷 = 𝐶𝐸, we have 𝐵𝐸 = 𝐶𝐷.

Also, because 𝐴𝐵 = 𝐴𝐶, and 𝐴𝐷 = 𝐴𝐸, then we obtain

△ 𝐴𝐵𝐸 ≌△ 𝐴𝐶𝐷（AAS）.

B. Side-Angle-Side (SAS) Congruence

(Page 80)

1.（1） Yes, they are all congruent by SAS.

（2） Yes, they are congruent by SAS.

2. Proof. ∵ 𝐴𝑂 = 𝐵𝑂, ∠1 = ∠2, 𝑂𝑃 = 𝑂𝑃

∴ △ 𝐴𝑂𝑃 ≌△ 𝐵𝑂𝑃（𝑆𝐴𝑆）

3. Solution. ∵ 𝐴𝐸 = 𝐷𝐵

∴ 𝐴𝐸 + 𝐵𝐸 = 𝐷𝐵 + 𝐵𝐸

∴ 𝐴𝐵 = 𝐷𝐸

∵ 𝐵𝐶 ∥ 𝐸𝐹

∴ ∠𝐴𝐵𝐶 = ∠𝐷𝐸𝐹

∵ 𝐴𝐵 = 𝐷𝐸, ∠𝐴𝐵𝐶 = ∠𝐷𝐸𝐹, 𝐵𝐶 = 𝐸𝐹

∴ △ 𝐴𝐵𝐶 ≌△ 𝐷𝐸𝐹 (𝑆𝐴𝑆).

（第 2 题）

Page 13 of 26

C. Angle-Side-Angle (ASA) and Angle-Angle-Side (AAS) Congruence

(Page 83)

1. ∵ 𝐹𝐵 = 𝐶𝐸

∴ 𝐹𝐵 + 𝐹𝐶 = 𝐶𝐸 + 𝐹𝐶

∴ 𝐵𝐶 = 𝐸𝐹

∵ 𝐴𝐵 ∥ 𝐸𝐷, 𝐴𝐶 ∥ 𝐹𝐷

∴ ∠𝐵 = ∠𝐸, ∠𝐴𝐶𝐵 = ∠𝐷𝐹𝐸

∵ ∠𝐵 = ∠𝐸, 𝐵𝐶 = 𝐸𝐹, ∠𝐴𝐶𝐵 = ∠𝐷𝐹𝐸

∴ △ 𝐴𝐵𝐶 ≌△ 𝐷𝐸𝐹 (𝐴𝑆𝐴), and 𝐴𝐵 = 𝐷𝐸, 𝐴𝐶 = 𝐷𝐹.

2. Proof. ∵ 𝐵𝐸 ⊥ 𝐴𝐶, 𝐶𝐷 ⊥ 𝐴𝐵

∴ ∠𝐴𝐸𝐵 = ∠𝐴𝐷𝐶 = 90°

∵ ∠𝐴𝐸𝐵 = ∠𝐴𝐷𝐶, ∠𝐴 = ∠𝐴, 𝐴𝐵 = 𝐴𝐶

∴ △ 𝐴𝐸𝐵 ≌△ 𝐴𝐷𝐶 (𝐴𝐴𝑆)

∴ 𝐵𝐸 = 𝐶𝐷

D. Hypotenuse-Leg (HL) Congruence

(Page 85)

1. Proof.

∵ 𝐷𝐸 ⊥ 𝐴𝐵, 𝐷𝐹 ⊥ 𝐴𝐶

∴ ∠𝐷𝐸𝐵 = ∠𝐷𝐹𝐶 = 90°

∵ 𝐵𝐷 = 𝐶𝐷, 𝐷𝐸 = 𝐷𝐹

∴ △ 𝐵𝐸𝐷 ≌△ 𝐶𝐹𝐷 (𝐻𝐿).

Page 14 of 26

2. ∵ ∠𝐶 = ∠𝐷 = 90°

∴ △ 𝐴𝐵𝐶 and △ 𝐴𝐵𝐷 are right triangles.

∵ 𝐴𝐶＝𝐴𝐷, 𝐴𝐵 = 𝐴𝐵

∴ △ 𝐴𝐵𝐶 ≌△ 𝐴𝐵𝐷 (𝐻𝐿)

∴ 𝐵𝐶 = 𝐵𝐷.

3. ∵ 𝑃𝐶 ⊥ 𝑂𝐴, 𝑃𝐷 ⊥ 𝑂𝐵,

∴ ∠𝑂𝐶𝑃 = ∠𝑂𝐷𝑃 = 90°

∴ △ 𝑂𝐶𝑃 and △ 𝑂𝐷𝑃 are right triangles.

∵ 𝑃𝐶 = 𝑃𝐷, 𝑂𝑃 = 𝑂𝑃

∴ △ 𝑂𝐶𝑃 ≌△ 𝑂𝐷𝑃 (𝐻𝐿)

∴ ∠1 = ∠2.

4. Proof. ∵ AD is the height.

∴ ∠𝐴𝐷𝐵 = ∠𝐴𝐷𝐶 = 90°

∴ △ 𝐴𝐷𝐵 and △ 𝐴𝐷𝐶 are right triangles.

∵ 𝐴𝐵 = 𝐴𝐶, 𝐴𝐷 = 𝐴𝐷

∴ △ 𝐴𝐷𝐵 ≌△ 𝐴𝐷𝐶 (𝐻𝐿)

∴ 𝐵𝐷 = 𝐶𝐷, ∠𝐵𝐴𝐷 = ∠𝐶𝐴𝐷.

3.4.3 Angle Bisector

A. Compass and Straightedge Construction on Angle Bisector

(Page 86)

1. ∠𝐷𝐴𝐶 is ∠𝐵, as shown in the graph below.

Page 15 of 26

2. Please see the graph below.

B. Properties and Criteria of Angle Bisector

(Page 88)

Solution. ∵ AD is an angle bisector, 𝐷𝐸 ⊥ 𝐴𝐵, and 𝐷𝐹 ⊥ 𝐴𝐶

∴ 𝐷𝐸 = 𝐷𝐹, ∠𝐷𝐸𝐵 = ∠𝐷𝐹𝐶 = 90°.

∵ 𝐵𝐷 = 𝐶𝐷, 𝐷𝐸 = 𝐷𝐹

∴ △ 𝐷𝐸𝐵 ≌△ 𝐷𝐹𝐶 (𝐻𝐿)

∴ 𝐸𝐵 = 𝐹𝐶.

C. Applications to Angle Bisector’s Properties and Criteria

(Page 90)

1. Proof.

∵ AD bisects ∠𝐵𝐴𝐶, 𝐵𝐹 ⊥ 𝐴𝐶, and 𝐶𝐸 ⊥ 𝐴𝐵.

∴ 𝐷𝐸 = 𝐷𝐹, ∠𝐷𝐸𝐵 = ∠𝐷𝐹𝐶 = 90°

∵ ∠𝐷𝐸𝐵 = ∠𝐷𝐹𝐶, 𝐷𝐸 = 𝐷𝐹, ∠𝐸𝐷𝐵 = ∠𝐹𝐷𝐶

∴ △ 𝐷𝐸𝐵 ≌△ 𝐷𝐹𝐶 (𝐴𝑆𝐴), and so 𝐵𝐷 = 𝐶𝐷.

2. Proof. Connect AP.

Page 16 of 26

∵ 𝑃𝐸 ⊥ 𝐴𝐵，𝑃𝐹 ⊥ 𝐴𝐶

∴ ∠𝑃𝐸𝐴 = ∠𝑃𝐹𝐴 = 90°

∵ 𝐴𝐸 = 𝐴𝐹，𝐴𝑃 = 𝐴𝑃

∴ △ 𝑃𝐸𝐴 ≌△ 𝑃𝐹𝐴 (𝐻𝐿)

∴ ∠𝑃𝐴𝐸 = ∠𝑃𝐴𝐹

∴ Point P is on the angle bisector of ∠𝐵𝐴𝐶.

3. Proof. ∵ 𝐵𝐹 ⊥ 𝐴𝐶，𝐶𝐸 ⊥ 𝐴𝐵

∴ ∠𝐷𝐸𝐵 = ∠𝐷𝐹𝐶 = 90°

∵ ∠𝐸𝐷𝐵 = ∠𝐹𝐷𝐶, ∠𝐷𝐸𝐵 = ∠𝐷𝐹𝐶, 𝐵𝐸 = 𝐶𝐹

∴ △ 𝐵𝐸𝐷 ≌△ 𝐶𝐹𝐷 (𝐴𝐴𝑆)

∴ 𝐷𝐸 = 𝐷𝐹

Also, ∵ 𝐵𝐹 ⊥ 𝐴𝐶，𝐶𝐸 ⊥ 𝐴𝐵

∴ AD bisects ∠𝐵𝐴𝐶.

3.4.4 Perpendicular Bisector

A. Compass and Straightedge Construction on Perpendicular Bisector

(Page 91)

1． 2．

Page 17 of 26

B. Properties and Criteria of Perpendicular Bisector

(Page 93)

Proof. ∵ ∠𝐷𝐸𝐴 = ∠𝐶𝐸𝐵, 𝐷𝐸 = 𝐶𝐸, ∠𝐶 = ∠𝐷

∴ △ 𝐴𝐷𝐸 ≌△ 𝐵𝐶𝐸 (𝐴𝑆𝐴)

∴ 𝐸𝐴 = 𝐸𝐵

∴ Point E is on the perpendicular bisector of AB.

C. Applications to Perpendicular Bisector’s Properties and Criteria

(Page 95)

1. 𝐴𝐶 = 6𝑐𝑚

2. Proof. ∵ 𝐴𝐵 = 𝐴𝐷, 𝐵𝐶 = 𝐶𝐷

∴ AC perpendicularly bisects BD.

∴ 𝐵𝑃 = 𝐷𝑃.

3.4.5 Reflection Transformation

A. Drawing of Reflection Figure and Its Properties

(Page 97)

1. 2.

B. Properties of Point Symmetry in Coordinate System

(Page 98)

1.

Page 18 of 26

𝐴(1 , 3) 𝐵(−1, 2) 𝐶(4, −5) 𝐷(3, 1) 𝐸(0, 4)

（ 1 , -3 ） （ -1 , -2 ） （ 4 , -5 ） （ 3 , -1 ） （ 0 , -4 ）

（ -1 , 3 ） （ 1 , 2 ） （ -4 , -5 ） （ -3 , 1 ） （ 0 , 4 ）

2.（1）

（2）𝐴’(2, 3), 𝐵’(3, 1), 𝐶’(−1, −2)

3.5 Isosceles Triangles

3.5.1 Properties of Isosceles Triangles

(Page 100)

1. ∠𝐵 = ∠𝐶 = 30°

2.（1）∠𝐴𝐵𝐷 = 20°

（2）Proof. ∵𝐴𝐵 = 𝐴𝐷

∴ ∠𝐴𝐵𝐷 = ∠𝐴𝐷𝐵

∵ 𝐴𝐷 ∥ 𝐵𝐶

∴ ∠𝐴𝐷𝐵 = ∠𝐷𝐵𝐶

∴ ∠𝐴𝐵𝐷 = ∠𝐷𝐵𝐶

∴ BD bisects ∠𝐴𝐵𝐶.

3. （1）∠𝐵𝐴𝐶 = 120° （2）𝐵𝐷 = 6.

Page 19 of 26

3.5.2 Criteria for Isosceles Triangles

(Page 102)

1. Proof. ∵ ∠𝐶𝐴𝐷 = ∠𝐶 + ∠𝐵

∴ ∠𝐵 = 100° − 50° = 50°

∴ ∠𝐵 = ∠𝐶

∴ 𝐴𝐵 = 𝐴𝐶

∴ △ 𝐴𝐵𝐶 is an isosceles triangle.

2. △ 𝐶𝐸𝐵 is an isosceles triangle.

Proof. ∵ 𝐶𝐸 ∥ 𝐷𝐴

∴ ∠𝐶𝐸𝐵 = ∠𝐴

∵ ∠𝐴 = ∠𝐵

∴ ∠𝐶𝐸𝐵 = ∠𝐵

∴ 𝐶𝐸 = 𝐶𝐵

Thus, △ 𝐶𝐸𝐵 is an isosceles triangle.

3. Proof. ∵ ∠1 = ∠2

∴ 𝐸𝐵 = 𝐸𝐴

∵ 𝐷𝐸 = 𝐸𝐶, ∠1 = ∠2, 𝐸𝐵 = 𝐸𝐴

∴ △ 𝐵𝐷𝐸 ≌ △ 𝐴𝐶𝐸 (𝑆𝐴𝑆)

3.5.3 Equilateral Triangles

A. Properties and Criteria of Equilateral Triangles

(Page 104)

Page 20 of 26

1. 𝐵𝐶 = 10𝑐𝑚. 2. ∠𝐵𝐴𝐷 = 30°.

B. Properties and Applications of Right Triangles with a 𝟑𝟎° Angle

(Page 106)

1. 𝐴𝐵 = 20, 𝐷𝐸 = 5.

2. Proof. ∵ ∠𝐴 = 30°, ∠𝐴𝐶𝐵 = 90°

∴ ∠𝐴𝐵𝐶 = 60°

∵ BD bisects ∠𝐴𝐵𝐶.

∴ ∠𝐷𝐵𝐶 = ∠𝐷𝐵𝐴 =2

1∠𝐴𝐵𝐶 = 30°

∴ 𝐵𝐷 = 2𝐶𝐷

∵ ∠𝐷𝐴𝐵 = ∠𝐴 = 30°

∴ 𝐴𝐷 = 𝐵𝐷

∴ 𝐴𝐷 = 2𝐷𝐶

Page 21 of 26

Chapter 4 Pythagorean Theorem

4.1 Pythagorean Theorem

A. Pythagorean Theorem and Its Representation

(Page 112)

1. B 2. 12𝑐𝑚 3. 𝐴𝐵 = 12𝑐𝑚 4. 2 + 22 𝑐𝑚

5．𝐶𝐷 = 2.4𝑐𝑚.

6. The area is 12.5, and the perimeter is 231353 ++ ．

B. Application to Pythagorean Theorem

(Page 115)

1. 62

2. The chopstick is 13𝑐𝑚 long, and the cup is 12𝑐𝑚 tall.

3. 𝐴𝐵 = 12𝑚.

C. Representing Irrational Numbers on Number Line Using

Pythagorean Theorem

(Page 118)

1．√5

2. （1）

（2）

Page 22 of 26

（3）

3.

4.2 Converse Theorem of the Pythagorean Theorem

(Page 122)

1.（1）Yes. The side with length of 20.

（2）No. （3）No.

2.（1）No.

（2）Yes. The side with length of 13.

3. ∠𝐵 = 60°.

4.3 Applications to Pythagorean Theorem and Its Converse

A. Applications to Pythagorean Triple

(Page 123)

Yes, it is a Pythagorean triple.

B. Applications to Special Triangles

(Page 125)

1. The area of △ 𝐴𝐵𝐶 is 48𝑐𝑚².

2. The area of quadrilateral ABCD is 1+ 5．

Page 23 of 26

C. Applications of Rectangular Coordinate System

(Page 127)

1. The length of AB is 5.

2. The distance is √13.

3. The distance is 25 .

4.4 Rotation Transformation

A. Definition of Rotation and Its Properties

(Page 130)

1. The given graph is generated by an arrow rotating about the center in the

graph for 8 times with 45 degrees each time．

2.

3. After 4 rotations.

4. Point A is the center of rotation. The degree of rotation is 45 degrees.

Page 24 of 26

B. Reasoning and Calculation on Rotating 𝟗𝟎° on a Triangle About a

Point

(Page 132)

1. (1) Point A is the center of rotation.

(2) 90 degrees

(3) 𝐸𝐹 = 2√2.

2. 𝐴′(−4, 1).

Chapter 5 Statistics-Organizing and Handling Data

5.1 Arithmetic Mean and Weighted Mean

5.1.1 Meaning of Arithmetic Mean

(Page 137)

1. 51𝑘𝑔 2. 𝑎 = 2 3. 5.3 hours 4. 175.5

5.1.2 Weighted Mean

(Page 139)

1. Correct. Because Mary purchased the same weight of each type.

If Mary purchased different weight on each type, then the method is

incorrect.

2. (1) Student A’s score:

85 × 20% + 83 × 30% + 90 × 50% = 86.9.

Student B’s score:

80 × 20% + 85 × 30% + 92 × 50% = 87.5.

So, Student B should win the position.

Page 25 of 26

（2）Student A’s score: 6.865

290183285

Student B’s score: 8.855

292185280

In this case, Student A should win the position.

5.1.3 Construction and Interpretation of Pie Charts

(Page 140)

1. January: 115° February: 144° March: 101°

2.

5.2 Applications to Mean, Median, and Mode

5.2.1 Median and Mode

(Page 143)

Mean: 4.88𝑘𝑔 Median: 4.85𝑘𝑔 Mode: 4.8𝑘𝑔

5.2.2 Proper Choice from Mean, Median, and Mode

(Page 145)

1.

Data Mean Median Mode

20， 20， 21， 24， 27， 30， 32 7

174 24 20

0， 2， 3， 4， 5， 5， 10 7

29 4 5

－2， 0， 3， 3， 3， 8 2.5 3 3

－6， －4， －2， 2， 4， 6 0 0 none

Page 26 of 26

2. 6

3. Median: 8 shots Mode: 8 shots

5.3 Range, Variance, and Standard Deviation

5.3.1 Measure of Dispersion for Discrete Data

5.3.2 Decision Making Based on Variance

(Page 149)

1. 2.5 2. B