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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 (High)Construction of Triangles with Secondary Information

### Construction of Triangles using Angle, Angle, Perimeter

In this page, a short and to-the-point overview of constructing triangles with angle-angle-perimeter is provided. Perimeter is a secondary parameter.

The reasoning on how to approach the problem and how the procedure works are provided.

click on the content to continue..

To construct a triangle, "angle-angle-perimeter" is provided. Which of the following is directly constructed using the given parameters?

• the triangle /_\ ABC
• Only a straight line, with length equivalent to the given perimeter. The given angles cannot be placed anywhere on the given straight line.
• Only a straight line, with length equivalent to the given perimeter. The given angles cannot be placed anywhere on the given straight line.

The answer is "Only a straight line"

To construct a triangle /_\ ABC with given "angle-angle-perimeter" /_A=60^@, /_B=30^@, and Perimeter=9cm.  •  Construct a line bar(PQ) measuring 9cm

This is directly constructed from the given parameters.

Point A is marked at an arbitrary position and locating B or C is constrained by the given two angles. The objective now is to locate the points A and B in the line bar(PQ) and point C such that the triangle has the given perimeter and the two angles.

To construct a triangle /_\ ABC with given "angle-angle-perimeter" /_A=60^@, /_B=30^@, and Perimeter=9cm.  •  Construct a line bar(PQ) measuring the perimeter 9cm.

•  Point A is visualized in an arbitrary position. It is not marked yet.

•  From the point A, the given angle /_A is used to construct vec(AD).

•  The position of point B is also visualized and the given angle /_B is used to construct vec(BE).

The point of intersection of vec(AD) and vec(BE) marks point C.

In this procedure, the positions of A and B are not known.

With given "angle-angle-perimeter" we constructed bar(PQ). The objective now is to locate the points A and B in the line bar(PQ) and point C.

Consider rays vec(rs), vec(vw), and vec(tu). All these rays are at the given angle /_B. These rays make triangles /_\ARS, /_\AVW, and /_\ATU. These triangles have the given two angles. Which of the following property is noted for triangles /_\ARS and /_\ATU?

• perimeter of /_\ARS is smaller than the given perimeter of triangle
• perimeter of /_\ATU is greater than the given perimeter of triangle
• both the above
• both the above

The answer is "both the above". It is noted that at only one position between the points A and Q, the point B is located such that perimeter of /_\ABC equals the given perimeter.

With given "angle-angle-perimeter" we constructed bar(PQ).

The points A, B, and C are visualized such that the triangle /_\ABC satisfies the given perimeter and angles. Which of the following is an useful observation that will help to locate the point C ?

• the line segments bar(PA) and bar(AC) are equal
• the line segments bar(QB) and bar(BC) are equal
• the triangles /_\PAC and /_\QBC are isosceles triangles
• all the above
• all the above

The answer is "all the above".

With given "angle-angle-perimeter" we constructed bar(PQ). The points A, B, and C are visualized such that the triangle /_\ABC satisfies the given perimeter and angles.

It is observed that /_\PAC and /_\QBC are isosceles.
Which of the following is an useful observation that will help to locate the point C ?

• the angle /_P is half of the given angle /_A
• the angle /_Q is half of the given angle /_B
• both the above
• both the above

The answer is "both the above".

With given "angle-angle-perimeter" we constructed bar(PQ). The points A, B, and C are visualized such that the triangle /_\ABC satisfies the given perimeter and angles. Since these points are not marked yet, these are not shown in the figure.

It is observed that

•  /_\PAC and /_\QBC are isosceles

•  the angle /_CPQ is half of the given angle /_A

•  the angle /_CQP is half of the given angle /_B To construct the half angles at point P and point Q, the given angles are constructed in /_EPQ and /_FQP and bisected. Which of the following marks the point C?

• the angle bisectors at P and Q intersect at point C
• the angle bisectors at P and Q intersect at point C
• the angle bisectors at P and Q are parallel
• both the above

The answer is "the angle bisectors at P and Q intersect at point C".

With given "angle-angle-perimeter" we constructed bar(PQ). The points A, B, and C are visualized such that the triangle /_\ABC satisfies the given perimeter and angles. Since these points are not marked yet, these are not shown in the figure.

Point C is marked by angle bisectors of angles P and Q.
Using the property that
/_\CPA and /_\CQB are isosceles triangles
By copying an angle or by constructing perpendicular bisector on the base, the points A and B are marked.
/_\ABC is constructed.

Construction of Angle-Angle-Perimeter of triangle : Visualize that at points A, B, and C, and the /_\ PAC /_\ QBCare isosceles triangles. Locate point C by angle bisectors at point P and Q. Locate points A and B by either copying an angle or drawing perpendicular bisector on the base.

slide-show version coming soon