4 1 radicals and pythagorean theorem-x

Radicals and Pythagorean Theorem

Radicals and Pythagorean TheoremFrom here on we need a scientific calculator. You may use the digital calculators on your personal devices.

Radicals and Pythagorean TheoremFrom here on we need a scientific calculator. You may use the digital calculators on your personal devices.Make sure the calculator could displace your input so that you may check the input before executing it.


Square Root

From here on we need a scientific calculator. You may use the digital calculators on your personal devices.Make sure the calculator could displace your input so that you may check the input before executing it.

“9 is the square of 3” may be rephrased backwards as “3 is the square root of 9”.


Square Root




Definition: If a is > 0, and a2 = x, then we say a is the square root of x. This is written as sqrt(x) = a, or x = a.

Square Root



Example A.

a. Sqrt(16) =c.–3 =



b. 1/9 =d. 3 =

Square Root



Example A.

a. Sqrt(16) = 4c.–3 =



b. 1/9 =d. 3 =

Square Root



Example A.

a. Sqrt(16) = 4c.–3 =



b. 1/9 = 1/3d. 3 =

Square Root



Example A.

a. Sqrt(16) = 4c.–3 = doesn’t exist



b. 1/9 = 1/3d. 3 =

Square Root



Example A.




b. 1/9 = 1/3d. 3 = 1.732.. (calculator)

Square Root



Example A.




b. 1/9 = 1/3d. 3 = 1.732.. (calculator)

Note that the square of both +3 and –3 is 9, but we designate sqrt(9) or 9 to be +3.

Square Root



Example A.




b. 1/9 = 1/3d. 3 = 1.732.. (calculator)

Note that the square of both +3 and –3 is 9, but we designate sqrt(9) or 9 to be +3. We say “–3” is the “negative of the square root of 9”.

From here on we need a scientific calculator. You may use the digital calculators on your personal devices.Make sure the calculator could displace your input so that you may check the input before executing it.Square Root

0 02 = 0 0 = 0

1 12 = 1 1 = 1

2 22 = 4 4 = 2

3 32 = 9 9 = 3

4 42 = 16 16 = 4

5 52 = 25 25 = 5

6 62 = 36 36 = 6

7 72 = 49 49 = 7

8 82 = 64 64 = 8

9 92 = 81 81 = 9

10 102 = 100 100 = 10

11 112 = 121 121 = 11

Radicals and Pythagorean TheoremFollowing are the square numbers and square-roots that one needs to memorize.

0 02 = 0 0 = 0

1 12 = 1 1 = 1

2 22 = 4 4 = 2

3 32 = 9 9 = 3

4 42 = 16 16 = 4

5 52 = 25 25 = 5

6 62 = 36 36 = 6

7 72 = 49 49 = 7

8 82 = 64 64 = 8

9 92 = 81 81 = 9

10 102 = 100 100 = 10

11 112 = 121 121 = 11

Radicals and Pythagorean TheoremFollowing are the square numbers and square-roots that one needs to memorize. These numbers are special because many mathematics exercises utilize square numbers.

0 02 = 0 0 = 0

1 12 = 1 1 = 1

2 22 = 4 4 = 2

3 32 = 9 9 = 3

4 42 = 16 16 = 4

5 52 = 25 25 = 5

6 62 = 36 36 = 6

7 72 = 49 49 = 7

8 82 = 64 64 = 8

9 92 = 81 81 = 9

10 102 = 100 100 = 10

11 112 = 121 121 = 11

We may estimate the sqrt of other small numbers using this table.


0 02 = 0 0 = 0

1 12 = 1 1 = 1

2 22 = 4 4 = 2

3 32 = 9 9 = 3

4 42 = 16 16 = 4

5 52 = 25 25 = 5

6 62 = 36 36 = 6

7 72 = 49 49 = 7

8 82 = 64 64 = 8

9 92 = 81 81 = 9

10 102 = 100 100 = 10

11 112 = 121 121 = 11

We may estimate the sqrt of other small numbers using this table. For example, 25 < 30 < 36


0 02 = 0 0 = 0

1 12 = 1 1 = 1

2 22 = 4 4 = 2

3 32 = 9 9 = 3

4 42 = 16 16 = 4

5 52 = 25 25 = 5

6 62 = 36 36 = 6

7 72 = 49 49 = 7

8 82 = 64 64 = 8

9 92 = 81 81 = 9

10 102 = 100 100 = 10

11 112 = 121 121 = 11

We may estimate the sqrt of other small numbers using this table. For example, 25 < 30 < 36hence 25 < 30 <36


0 02 = 0 0 = 0

1 12 = 1 1 = 1

2 22 = 4 4 = 2

3 32 = 9 9 = 3

4 42 = 16 16 = 4

5 52 = 25 25 = 5

6 62 = 36 36 = 6

7 72 = 49 49 = 7

8 82 = 64 64 = 8

9 92 = 81 81 = 9

10 102 = 100 100 = 10

11 112 = 121 121 = 11

We may estimate the sqrt of other small numbers using this table. For example, 25 < 30 < 36hence 25 < 30 <36or 5 < 30 < 6


0 02 = 0 0 = 0

1 12 = 1 1 = 1

2 22 = 4 4 = 2

3 32 = 9 9 = 3

4 42 = 16 16 = 4

5 52 = 25 25 = 5

6 62 = 36 36 = 6

7 72 = 49 49 = 7

8 82 = 64 64 = 8

9 92 = 81 81 = 9

10 102 = 100 100 = 10

11 112 = 121 121 = 11

We may estimate the sqrt of other small numbers using this table. For example, 25 < 30 < 36hence 25 < 30 <36or 5 < 30 < 6Since 30 is about half way between 25 and 36,


0 02 = 0 0 = 0

1 12 = 1 1 = 1

2 22 = 4 4 = 2

3 32 = 9 9 = 3

4 42 = 16 16 = 4

5 52 = 25 25 = 5

6 62 = 36 36 = 6

7 72 = 49 49 = 7

8 82 = 64 64 = 8

9 92 = 81 81 = 9

10 102 = 100 100 = 10

11 112 = 121 121 = 11

We may estimate the sqrt of other small numbers using this table. For example, 25 < 30 < 36hence 25 < 30 <36or 5 < 30 < 6Since 30 is about half way between 25 and 36, so we estimate that30 5.5.


0 02 = 0 0 = 0

1 12 = 1 1 = 1

2 22 = 4 4 = 2

3 32 = 9 9 = 3

4 42 = 16 16 = 4

5 52 = 25 25 = 5

6 62 = 36 36 = 6

7 72 = 49 49 = 7

8 82 = 64 64 = 8

9 92 = 81 81 = 9

10 102 = 100 100 = 10

11 112 = 121 121 = 11

We may estimate the sqrt of other small numbers using this table. For example, 25 < 30 < 36hence 25 < 30 <36or 5 < 30 < 6Since 30 is about half way between 25 and 36, so we estimate that30 5.5.In fact 30 5.47722….


Equations of the form x2 = c has two answers:x = +c or –c if c>0.




(If c<0, there is no real solution.)



Example B. Solve the following equations.a. x2

= 25





= 25 x = ±25





= 25 x = ±25 = ±5





= 25 x = ±25 = ±5

b. x2 = –4





= 25 x = ±25 = ±5

b. x2 = –4 Solution does not exist.





= 25 x = ±25 = ±5


c. x2 = 8





= 25 x = ±25 = ±5


c. x2 = 8 x = ±8





= 25 x = ±25 = ±5


c. x2 = 8 x = ±8 ±2.8284.. by calculator





= 25 x = ±25 = ±5



exact answer approximate answer





= 25 x = ±25 = ±5



exact answer approximate answerSquare-roots numbers show up in geometry for measuring distances because of the Pythagorean Theorem.


A right triangle is a triangle with a right angle as one of its angle.


A right triangle is a triangle with a right angle as one of its angles. The longest side C of a right triangle is called the hypotenuse,


hypotenuseC

A right triangle is a triangle with a right angle as one of its angles. The longest side C of a right triangle is called the hypotenuse, the two sides A and B forming the right angle are called the legs.


hypotenuse

legsA

BC

A right triangle is a triangle with a right angle as one of its angles. The longest side C of a right triangle is called the hypotenuse, the two sides A and B forming the right angle are called the legs. Pythagorean TheoremGiven a right triangle with labeling as shown, then A2 + B2 = C2 as shown


hypotenuse

legsA

BC

Radicals and Pythagorean TheoremThere are two types of problems that use the Pythagorean Theorem to find the sides of the right triangles

Radicals and Pythagorean TheoremThere are two types of problems that use the Pythagorean Theorem to find the sides of the right triangles-finding the hypotenuse versus finding the legs.

Example C. Find the missing side of the following right triangles. Find the exact answer and the approximate answer. Draw.a. a = 5, b = 12, c = ?



Since it is a right triangle,122 + 52 = c2




144 + 25 = c2




144 + 25 = c2

169 = c2




144 + 25 = c2

169 = c2

So c = ±169 = ±13




144 + 25 = c2

169 = c2

So c = ±169 = ±13Since length can’t be negative, therefore c = 13.


b. a = 5, c = 12, b = ?Radicals and Pythagorean Theorem

b. a = 5, c = 12, b = ?Since it is a right triangle,b2 + 52 = 122



b2 + 25 = 144



b2 + 25 = 144b2 = 144 – 25



b2 + 25 = 144b2 = 144 – 25 So b = ±119 ±10.9.



b2 + 25 = 144b2 = 144 – 25 So b = ±119 ±10.9. But length can’t be negative, therefore b = 119 10.9





The Distance Formula




Let (x1, y1) and (x2, y2) be two points, D = distance between them,


(x1, y1)

(x2, y2)

D




Let (x1, y1) and (x2, y2) be two points, D = distance between them, then D2 = Δx2 + Δy2


(x1, y1)

(x2, y2)

Δy

Δx

D




Let (x1, y1) and (x2, y2) be two points, D = distance between them, then D2 = Δx2 + Δy2 where


(x1, y1)

(x2, y2)

Δx = x2 – x1

Δy

Δx = x2 – x1

D






(x1, y1)

(x2, y2)

Δx = x2 – x1 Δy = y2 – y1and

Δy = y2 – y1

Δx = x2 – x1

D

by the Pythagorean Theorem.






(x1, y1)

(x2, y2)

Δx = x2 – x1 Δy = y2 – y1and

Δy = y2 – y1

Δx = x2 – x1by the Pythagorean Theorem.Hence we’ve the Distant Formula:D = √ Δx2 + Δy2

D






(x1, y1)

(x2, y2)

Δx = x2 – x1 Δy = y2 – y1and

Δy = y2 – y1

Δx = x2 – x1by the Pythagorean Theorem.Hence we’ve the Distant Formula:D = √ Δx2 + Δy2 = √ (x2 – x1)2 + (y2 – y1)2

D

Example D. Find the distance between (–1, 3) and (2, –4). (-1, 3)– ( 2, -4)

D


–3, 7

Δx Δy


D

7

–3


–3, 7

Δx Δy


D = (–3)2 + 72 = 58 7.62

D

7

–3


–3, 7

Δx Δy


D = (–3)2 + 72 = 58 7.62

D

7

–3


–3, 7

Δx Δy

Higher Root


D = (–3)2 + 72 = 58 7.62

D

7

–3


–3, 7

Δx Δy

Higher RootIf r3 = x, then we say a is the cube root of x.


D = (–3)2 + 72 = 58 7.62

D

7

–3


–3, 7

Δx Δy

Higher RootIf r3 = x, then we say a is the cube root of x. We write this as x = r. 3


D = (–3)2 + 72 = 58 7.62

D

7

–3


–3, 7

Δx Δy

Higher RootIf r3 = x, then we say a is the cube root of x. We write this as x = r. In general, if r k = x, then we say r is the k’th root of x, 3


D = (–3)2 + 72 = 58 7.62

D

7

–3


–3, 7

Δx Δy

Higher RootIf r3 = x, then we say a is the cube root of x. We write this as x = r. In general, if r k = x, then we say r is the k’th root of x, and we write it as a = x.

3

k


D = (–3)2 + 72 = 58 7.62

D

7

–3


–3, 7

Δx Δy

Higher RootIf r3 = x, then we say a is the cube root of x. We write this as x = r. In general, if r k = x, then we say r is the k’th root of x, and we write it as a = x. In the cases of even roots, i.e. k = 2, 4, 6, … we must have x > 0 and that x = a > 0.

3

k

k


D = (–3)2 + 72 = 58 7.62

D

7

–3


–3, 7

Δx Δy

3

k

k

Example E.

a. 8 =3



D = (–3)2 + 72 = 58 7.62

D

7

–3


–3, 7

Δx Δy

3

k

k

Example E.

a. 8 = 23



D = (–3)2 + 72 = 58 7.62

D

7

–3


–3, 7

Δx Δy

3

k

k

Example E.

a. 8 = 2 b. –1 =3 3



D = (–3)2 + 72 = 58 7.62

D

7

–3


–3, 7

Δx Δy

3

k

k

Example E.

a. 8 = 2 b. –1 = –13 3



D = (–3)2 + 72 = 58 7.62

D

7

–3


–3, 7

Δx Δy

3

k

k

Example E.

a. 8 = 2 b. –1 = –1 c.–27 =3 3 3



D = (–3)2 + 72 = 58 7.62

D

7

–3


–3, 7

Δx Δy

3

k

k

Example E.

a. 8 = 2 b. –1 = –1 c.–27 = –3 3 3 3



D = (–3)2 + 72 = 58 7.62

D

7

–3


–3, 7

Δx Δy

3

k

k

Example E.

a. 8 = 2 b. –1 = –1 c.–27 = –3

d. 16 =

3 3 3


4


D = (–3)2 + 72 = 58 7.62

D

7

–3


–3, 7

Δx Δy

3

k

k

Example E.

a. 8 = 2 b. –1 = –1 c.–27 = –3

d. 16 = 2

3 3 3


4


D = (–3)2 + 72 = 58 7.62

D

7

–3


–3, 7

Δx Δy

3

k

k

Example E.

a. 8 = 2 b. –1 = –1 c.–27 = –3

d. 16 = 2 e. –16 =

3 3 3


4 4


D = (–3)2 + 72 = 58 7.62

D

7

–3


–3, 7

Δx Δy

3

k

k

Example E.

a. 8 = 2 b. –1 = –1 c.–27 = –3

d. 16 = 2 e. –16 = not real

3 3 3


4 4


D = (–3)2 + 72 = 58 7.62

D

7

–3


–3, 7

Δx Δy

3

k

k

Example E.

a. 8 = 2 b. –1 = –1 c.–27 = –3

d. 16 = 2 e. –16 = not real f. 10 ≈ 2.15..

3 3 3


4 4 3

Radicals and Pythagorean TheoremRational and Irrational Numbers

The number 2 is the length of the hypotenuse of the right triangle as shown.


2 = 12 + 12

1

1



2 = 12 + 12

1

1

It can be shown that 2 is not a fraction like 3/5, ½ etc..



2 = 12 + 12

1

1

It can be shown that 2 is not a fraction like 3/5, ½ etc.. i.e. 2 ≠ P/Q of any two integers P and Q.



2 = 12 + 12

1

1

It can be shown that 2 is not a fraction like 3/5, ½ etc.. i.e. 2 ≠ P/Q of any two integers P and Q.Therefore we said that2 is an irrational (non–ratio) number.



2 = 12 + 12

1

1

It can be shown that 2 is not a fraction like 3/5, ½ etc.. i.e. 2 ≠ P/Q of any two integers P and Q.Therefore we said that2 is an irrational (non–ratio) number. Most real numbers are irrational.



2 = 12 + 12

1

1


The real line is populated sparsely by fractional numbers. The Pythagorean school of the ancient Greeks had believed that all the measurable quantities in the universe are fractional quantities. The “discovery” of these extra irrational numbers caused a profound intellectual crisis.



2 = 12 + 12

1

1


The real line is populated sparsely by fractional numbers. The Pythagorean school of the ancient Greeks had believed that all the measurable quantities in the universe are fractional quantities. The “discovery” of these extra irrational numbers caused a profound intellectual crisis. It wasn’t until the last two centuries that mathematicians clarified the strange questions “How many and what kind of numbers are there?”

https://www.math.utah.edu/~pa/math/q1.html





Radicals and Pythagorean TheoremExercise A. Solve for x. Give both the exact and approximate answers. If the answer does not exist, state so. 1. x2 = 1 2. x2 – 5 = 4 3. x2 + 5 = 4 4. 2x2 = 31 5. 4x2 – 5 = 4 6. 5 = 3x2 + 1 7. 4x2 = 1 8. x2 – 32 = 42 9. x2 + 62 = 102

10. 2x2 + 7 = 11 11. 2x2 – 5 = 6 12. 4 = 3x2 + 5

x

3

4

Exercise B. Solve for x. Give both the exact and approximate answers. If the answer does not exist, state so.

13. 4

3

x14. x

12

515.

x

1

116. 2

1

x17. 3 2

3

x18.


x

4

19.

x

x20.

3 /3

21.43 5 2 6 /3

Exercise C. Given the following information, find the rise and run from A to B i.e. Δx and Δy. Find the distance from A to B.

A22.

B

A

23.

B

24. A = (2, –3) , B = (1, 5) 25. A = (1, 5) , B = (2, –3)26. A = (–2 , –5) , B = (3, –2) 27. A = (–4 , –1) , B = (2, –3)28. Why is the distance from A to B the same as from B to A?

Exercise D. Find the exact answer.Radicals and Pythagorean Theorem

3–129. 30. 13

–12534. 8 31. –13

83–2732.

33. –13

643

10035. 100036. 3

10,00037. 1,000,00038. 3

0.0139. 0.00140. 3

0.000141. 0.00000142. 3

4 1 radicals and pythagorean theorem-x

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