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Thought-Process to Discover Knowledge

Welcome to nubtrek.

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.
mathsConstruction / Practical Geometry (basics)Construction of Triangles (Fundamental Shape of Polygons)

### Construction of Triangles with Other information

In this page, a short analysis of constructing triangles with the following is provided.

•  angle-angle-side (equivalent to angle-side-angle)

•  angle-angle-angle (only 2 parameters)

•  side-side-angle (two possible triangles)

•  side-side-altitude (two possible triangles)

•  rightangle-angle-side (equivalent to angle-side-angle)

click on the content to continue..

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

Which of the following method can be used to construct?

• 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 a triangle, "angle-angle-angle" is provided.

Which of the following method can be used to construct?

• this is equivalently "side-side-side" and so the triangle can be constructed
• the given angles are not 3 independent parameters
• the given angles are not 3 independent parameters

The answer is "the given angles are not 3 independent parameters as given 2 angles the third angle is defined". The 3 angles together form 2 independent parameters.

To construct a triangle, "side-side-angle" is provided. Which of the following method can be used to construct?

• the given information "side-side-angle" may lead to two possible solutions
• the given information "side-side-angle" may lead to two possible solutions
• the given information can be rewritten as "side-angle-side" and a triangle may be constructed

The answer is "the given information "side-side-angle" may lead to two possible solutions"

To construct a triangle, "side1-side2-angle" are provided. The top portion of the figure illustrates the construction.

•  Side2 bar(AB) is constructed

•  /_A is constructed with vec(AP).

•  side1 is to be constructed. With a compass from point B, an arc is drawn. This cuts at 2 points C1 and C2. The triangles /_\ ABC1 and /_\ ABC2 both satisfy the given parameters

bar(AB) is the given side2

/_A is the given angle

Both bar(BC1) and bar(BC2) equal side1.

Since this leads to two possible solutions, the triangle is not uniquely defined. Under some conditions, the given information "side1-side2-angle" leads to an unique triangle. This is illustrated in the lower part of the figure.

•  when the given angle is obtuse, the side1 has to be greater than side2. This is illustrated in /_\ ABC3.

•  when the side1 is greater than the side2. This is illustrated in /_\ ABC4

To construct a triangle, "side-side-altitude" are provided. Which of the following method can be used to construct?

• the given information "side-side-altitude" may lead to two possible solutions
• the given information "side-side-altitude" may lead to two possible solutions
• the given information can be rewritten as "side-angle-side" and a triangle may be constructed

The answer is "the given information "side-side-altitude" may lead to two possible solutions"

To construct a triangle, "side1-side2-altitude" is provided. The figure illustrates possible construction.

•  Side2 bar(AB) is constructed

•  Parallel at altitude is constructed at point E.

•  Side1 is to be constructed. With a compass from point A, an arc is drawn. This cuts at 2 points C1 and C2. The triangle /_\ ABC1 and /_\ ABC2 both satisfy the given parameters

bar(AB) is the given side2

bar(DE) is the given altitude

Both bar(AC1) and bar(AC2) equal side1.

Since this leads to two possible solutions, the given S A L does not uniquely define a triangle.

To construct a triangle, "right angle-angle1-side" are provided. Which of the following method can be used to construct?

• it can be converted to "right angle-side-angle2" by sum of interior angle property
• it can be converted to "right angle-side-angle2" by sum of interior angle property
• an unique triangle cannot be constructed with the given info

The answer is "it can be converted to right angle-side-angle2 by sum of interior angle property"

To construct a triangle, "right angle-angle1-hypotenuse" is provided. Which of the following method can be used to construct?

• it can be converted to angle1-hypotenuse-angle2 by sum of interior angle property
• it can be converted to angle1-hypotenuse-angle2 by sum of interior angle property
• an unique triangle cannot be constructed with the given info

The answer is "it can be converted to angle1-hypotenuse-angle2 by sum of interior angle property"

slide-show version coming soon