__maths__>__Construction / 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`=9`cm. • Construct a line `bar(PQ)` measuring `9`cm

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`=9`cm.* • Construct a line `bar(PQ)` measuring the perimeter `9`cm.

• 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` `/_\ QBC`are 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*