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Thought-Process to Discover Knowledge

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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.

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mathsConstruction / Practical Geometry (basics)Construction of Quadrilaterals

Fundamentals of Construction : Quadrilaterals

In this page, a short overview of approaching construction problems for various quadrilateral forms is provided.

 •  A quadrilateral form has some additional properties, for example: all sides of a square are equal and the four angles are `90^@`.

 •  A quadrilateral is seen to be combination of two triangles.

Using the properties, the construction is simplified into combination of triangles of sss, sas, asa, rhs, sal.

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In practical geometry, we study construction of various figures using a scale, protractor, compass, and set-squares. Using these instruments, some fundamental elements of construction are realized.4 fundamental elements of practical geometry What are the fundamental elements of practical geometry?

  • constructing a line, an arc, an angle, and a parallel
  • constructing collinear points, equidistant points, equiangular points, and parallel points.
  • both the above
  • both the above

The answer is "both the above"

To construct any shape, the fundamental elements we use are 4 fundamental elements of practical geometry  •  Construct a line passing through the two given points using a ruler or scale (collinear points)

 •  Construct a ray at the given angle using a protractor (equiangular points)

 •  Construct an arc at the given distance using a compass and a ruler or scale (equidistant points)

 •  construct a parallel line using a set-square and a scale or another set-square. (points on a parallel)

Triangle `/_\ ABC` has the following measurements as the parameters

 •  side `bar(AB)`

 •  side `bar(BC)`

 •  side `bar(CA)`

 •  angle `/_A`

 •  angle `/_B`

 •  angle `/_C`

 •  altitude `bar(HC)`

Note that the three angles are not truly independent as `/_A + /_B + /_C = 180^@`

These are `7` parameters in total.triangles introduction 1 How many independent parameters define a triangle?

  • `7`
  • `3`
  • `3`

The answer is "`3`". `3` independent parameters define a triangle. Other `3` can be derived from the given three parameters.

Considering that triangles are defined by three independent parameters, one of the following possible combination is provided to construct a triangle.

 •  Side-Side-Side

 •  Side-Angle-Side

 •  Angle-Side-Angle

 •  right angle-side-hypotenuse

 •  angle-side-altitude

angle-angle-angle is just two independent parameters as the sum of angles are `180^@`.

side-angle-angle is same as angle-side-angle, as the sum of angle property defines the third angle.
triangles introduction 1 The given parameters have to satisfy properties of a triangle.

eg1: `4,4,10` cm cannot be sss of a triangle, as sum of any two sides of a triangle has to be greater than the third side.

eg2: `200^@` cannot be an interior angle of a triangle, as sum of angles has to be `180^@`

To construct a triangle, "angle-angle-side" is provided.

Which of the following method can be used to construct the triangle?

  • this is equivalently "angle-side-angle" as sum of angle is `180^@`
  • this is equivalently "angle-side-angle" as sum of angle is `180^@`
  • the given "angle-angle-side" cannot be used to construct a triangle

The answer is "this is equivalently "angle-side-angle" as sum of angle is `180^@`".

To construct triangles, sometimes the properties of triangles are used. Similarly, to construct quadrilaterals, the properties of the quadrilaterals are used.

What are quadrilaterals?

  • plane figures of `4` sides
  • plane figures of `4` sides
  • plane figures of `8` sides

The answer is "plane figures of `4` sides"

Types of Quadrilaterals

 •  Irregular Quadrilateral

 •  Parallelogram

 •  Rhombus

 •  Rectangle

 •  Square

 •  Trapezium

 •  Kite

The possible parameters defining a quadrilateral are

 •  the four sides `bar(AB)`, `bar(BC)`, `bar(CD)`, and `bar(AD)`

 •  the four angles `/_A`, `/_B`, `/_C`, and `/_D`

 •  the two diagonals `bar(AC)` and `bar(BD)` quadrilateral introduction How many independent parameters define a quadrilateral?

  • all `10`
  • only `5`
  • only `5`

The answer is "only `5`"

Consider the quadrilateral as four vertices. The four vertices make two triangles. Quadrilateral is considered to be made of two triangles. Three vertices form the first triangle. The fourth vertex is placed in reference to the first triangle and thus, the fourth vertex and two more vertices of the first triangle form the second triangle.

 •  `/_\ ABC` and `/_\ ACD` define the four vertices

 •  `/_\ ABC` and `/_\ ABD` define the four vertices

 •  `/_\ ABC` and `/_\ BCD` define the four verticesquadrilateral introductionA quadrilateral is defined by `5` parameters.

The five parameters defining the quadrilateral are considered as two triangles and the various methods of constructing triangles are employed.quadrilateral introduction  •  Construction of Triangle SSS

 •  Construction of Triangle SAS

 •  Construction of Triangle ASA

 •  Construction of Triangle RHS

 •  Construction of Triangle SAL

Construction of quadrilateral has very many possible formulations. No need to memorize or learn by rote.

Always look for triangles, that is one point is defined in reference to another two points by one of `SSS`, `SAS`, `ASA`, `RHS`, or `SAL` methods.

Look for properties of specific type of quadrilaterals to figure out more information.

This is the most important learning in this topic. Once you work out the first problem in this formulation, the rest of the problems will become very very easy.

Construction of Quadrilateral : The `4` vertices of quadrilateral are defined by `5` parameters. The `5` parameters are considered as two triangles.quadrilateral introduction Once the two triangles are identified, use the SSS / SAS / ASA / RHS / SAL methods to construct the vertices of the two triangles and thus, the quadrilateral.

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