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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

•  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.

click on the content to continue..

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. 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  •  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. 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.
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.

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

The answer is "plane figures of 4 sides"

•  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) 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 verticesA 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.  •  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.

Construction of Quadrilateral : The 4 vertices of quadrilateral are defined by 5 parameters. The 5 parameters are considered as two triangles. 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.