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Thought-Process to Discover Knowledge

Welcome to **nub****trek**.

Books and other education websites provide "matter-of-fact" knowledge. Instead, nubtrek provides a thought-process to discover knowledge.

In each of the topic, the outline of the thought-process, for that topic, is provided for learners and educators.

Read in the blogs more about the unique learning experience at nubtrek.continue

Books and other education websites provide "matter-of-fact" knowledge. Instead, nubtrek provides a thought-process to discover knowledge.

In each of the topic, the outline of the thought-process, for that topic, is provided for learners and educators.

Read in the blogs more about the unique learning experience at nubtrek.continue

Welcome to **nub****trek**.

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think,

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learn.

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The content is presented in small-focused learning units to enable you to

think,

figure-out, &

learn.

To make best use of nubtrek, understand what is available.

nubtrek is designed to explain mathematics and science for young readers. Every topic consists of four sections.

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jogger,

exercise.

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Triangles Revised

» Congruent Triangles: Triangles that are identical.

» Similar Triangles: Triangles that are scaled.

» Right-Triangle: A triangle with one angle `90^@`.

» **3** independent parameters in a **triangle**

→ angle-side-angle

→ side-side-side

→ side-angle-side

→ angle-angle-side

→ (angle-angle-angle are not independent)

→ (side-side-angle leads to 2 possibilities

» **2** independent parameters in a **right-triangle**

→ side-angle

→ side-side

» **1** independent parameter in "class of similar-right-triangles"

→ angle

*plain and simple summary*

nub

*plain and simple summary*

nub

dummy

Basics of Triangles to learn trigonometry :

• Congruent and Similar Triangles.

• three parameters define a triangle

• two parameters define a right angled triangle

• one parameter define set of similar right angled triangles : angle

*simple steps to build the foundation*

trek

*simple steps to build the foundation*

trek

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In this page, the basics of triangles required to understand trigonometry is revised.

Starting on learning "Basics of Triangles". ;;In this page, the basics of triangles required to understand trigonometry is revised.

What is the sum of interior angles of a triangle?

- `90^@`
- `180^@`
- `270^@`
- `360^@`

The answer is '`180^@`'

What are "Congruent Triangles"?

- Triangles that are identical
- Triangles that exactly fit over
- both the above

The answer is 'both the above'. Note triangles can be flipped or rotated while comparing.

What are "Similar Triangles"?

- Triangles that have same shape
- Triangles that are scaled up or down version of one another
- both the above

The answer is 'both the above'

What is a right angled triangle?

- A triangle with one angle `90^@`
- A triangle with all angles equal.
- both the above

The answer is 'A triangle with one angle `90^@`'

How many independent parameters define a triangle?

- `1`
- `2`
- `3`
- `4`

The answer is '`3`'.

For example, given that the sides of a triangle are `1`, `1`, and the angle between the sides is `60^@`, it is derived that the triangle is a equilateral triangle. The independent parameters are given

• side-side-side

• side-angle-side

• angle-side-angle

• angle-angle-side

Note that, angle-angle-angle are not independent parameters as sum of angles in a triangle is `180^@`. So given two angles, third angle can be calculated.

How many independent parameters define a right angled triangle?

- `1`
- `2`
- `3`
- `4`

The answer is '`2`'.

For a right angled triangle, one angle is given as `90^@`, so the independent parameters are given

• side-side

• side-angle

What does "two independent parameters define a right angled triangle" mean?

- If two of 2 angles (excluding the right angle) or 3 sides are given, then all other angle or sides are exactly defined and mathematically computed.
- Two independent parameters define set of similar triangles.

The answer is "If two of 2 angles (excluding the right angle) or 3 sides are given, then all other angle or sides are exactly defined and mathematically computed".

The two arms to the right angle are given as `a` and `b`. Since two independent parameters define a right angled triangle, what is the third side `c`?

- `c = sqrt(a^2+b^2)`
- can not be computed

The answer is "`c = sqrt(a^2+b^2)`". This is the Pythagoras theorem.

** Pythagoras Theorem: ** It is understood that the two arms of right angled triangle defines the hypotenuse. As length of one arm changes, the hypotenuse also changes accordingly. The relationship between the arms and the hypotenuse is given by `c^2 = a^2+b^2`.

This happens to be a cute expression and it is used often in mathematics. Thus, it is one of the most popular theorems. The explanations like `c^2` represents area of square with hypotenuse as side etc. are proofs or analogies of this relationship.

The expression, in absolute sense, captures the relationship between two arms and the hypotenuse, as only two parameters are independent.

** Pythagoras Theorem (defined in first principles): ** A right angled triangle is defined by two independent parameters. Hence, The three parameters, hypotenuse and the two legs, are related. The relationship is given by the expression

`c^2 = a^2+b^2`

where `c` is the hypotenuse and `a` and `b` are the two legs.

How many independent parameters define the set of similar right angled triangles? (Figure shows three triangles, solid line, dotted line, and shaded.)

- `1`
- `2`
- `3`
- `4`

The answer is '`1`'.

Set of similar right angled triangles is defined by one angle.

*comprehensive information for quick review*

Jogger

*comprehensive information for quick review*

Jogger

dummy

**Congruent Triangles: ** Triangles that are identical.**Similar Triangles: ** Triangles that are scaled and have same shape.**Right Angled Triangle: ** Triangles having one angle as `90^@`.**Number of Parameters Defining a Triangle: ** Minimum three independent parameters are required to specify a triangle: SSS, SAS, ASA, AAS**Number of Parameters Defining a Right Angled Triangle: ** Minimum two independent parameters are required to specify a right angled triangle : SA or SS**Number of Parameters Defining set of Similar Right Angled Triangles: **One parameter define set of similar right angled triangles : A

*practice questions to master the knowledge*

Exercise

*practice questions to master the knowledge*

Exercise

*your progress details*

Progress

*About you*

Progress

What is the sum of interior angles of a triangle?

90

90 degree

180

180 degree

270

270 degree

360

360 degree

The answer is 180 degree.

What are "Congruent Triangles"?

identical

Triangles that are identical

exactly fit

Triangles that exactly fit over

above

both the above

The answer is 'both the above'. Note triangles can be flipped or rotated while comparing.

What are "Similar Triangles"?

same shape

Triangles that have same shape

scaled

Triangles that are scaled up or down version of one another

above

both the above

The answer is 'both the above'

What is a right angled triangle?

degree

A triangle with one angle 90 degree

equal

A triangle with all angles equal.

above

both the above

The answer is; A triangle with one angle 90 degree.

How many independent parameters define a triangle?

1

1

2

2

3

3

4

4

The answer is 3. ;; For example, given that the sides of a triangle are equal and the angle between the sides is 60 degree, it is derived that the triangle is a equilateral triangle. The independent parameters are given;; side-side-side ;; side-angle-side ;; angle-side-angle ;; angle-angle-side ;; Note that, angle-angle-angle are not independent parameters as sum of angles in a triangle is 180 degree. So given two angles, third angle can be calculated.

How many independent parameters define a right angled triangle?

1

1

2

2

3

3

4

4

The answer is 2. For a right angled triangle, one angle is given as 90 degree, so the independent parameters are given as;; side-side ;; side-angle

What does "two independent parameters define a right angled triangle" mean?

sides;angles;3;all;angle;exactly;mathematically;computed

If two of 2 angles (excluding the right angle) or 3 sides are given, then all other angle or sides are exactly defined and mathematically computed.

independent;parameters;similar;set

Two independent parameters define set of similar triangles.

The answer is "If two of 2 angles (excluding the right angle) or 3 sides are given, then all other angle or sides are exactly defined and mathematically computed".

The two arms to the right angle are given as a and b . Since two independent parameters define a right angled triangle, what is the third side c ?

see;sea;c;C;square root;a;A;square;bee;b;B

c equals square root of A squared plus B squared

cannot;can;not;compute

can not be computed

The answer is c equals square root of A squared plus B squared

Pythagoras Theorem: It is understood that the two arms of right angled triangle defines the hypotenuse. As length of one arm changes, the hypotenuse also changes accordingly. The relationship between the arms and the hypotenuse is given by c squared equals, a squared plus b squared. ;; This happens to be a cute expression and it is used often in mathematics. Thus, it is one of the most popular theorems. The explanations like c squared represents area etc. are just proofs of this relationship. ;; The expression, in absolute sense, captures the relationship between two arms and the hypotenuse, as only two parameters are independent.

Pythagoras Theorem: A right angled triangle is defined by two independent parameters. Hence, The three parameters, hypotenuse and the two legs, are related. The relationship is given by the expression c squared equals, a squared plus b squared; where c is the hypotenuse and a and b are the two legs.

How many independent parameters define the set of similar right angled triangles?

1

1

2

2

3

3

4

4

The answer is 1. ;; Set of similar right angled triangles is defined by one angle.

The following basics of triangles are required to learn trigonometry. : ;; Congruent and Similar Triangles. ;; three parameters define a triangle ;; two parameters define a right angled triangle ;; one parameter define set of similar right angled triangles : angle

The reference material is given for you to revise anytime.