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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.

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mathsConstruction / Practical Geometry (High)Construction of Triangles with Secondary Information

Construction of Triangles using Angle, Side, Sum of 2 Sides

In this page, a short and to-the-point overview of constructing triangles with angle-side-sum of two sides is provided. Sum of two sides 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, "side-angle-sum of other two sides" is provided. construction of triangle SA sum of 2 sides Which of the following is directly constructed using the given parameters?

  • the triangle `/_\ ABC`
  • the given angle with two line segments of given side and sum of sides
  • the given angle with two line segments of given side and sum of sides

The answer is "the given angle with two line segments of given side and sum of sides"

To construct a triangle `/_\ ABC` with given "side-angle-sum of other two sides" `bar(AB)=4`cm, `/_A=50^@`, and `bar(AC)+bar(BC) = 6`cm. construction of triangle SA sum of 2 sides  •  Construct a line and mark side `bar(AB)` with a compass measuring `4`cm

 •  measure `50^@` angle and draw a ray from point `A`

 •  measure `bar(AC)+bar(BC)` `=6`cm in a compass and draw an arc cutting the ray at `D`

This is directly constructed from the given parameters.

With given "side-angle-sum of two sides" we constructed `bar(AB)`, and `bar(AD)=bar(AC)+bar(CB)` at the given angle `/_A`. The objective now is to locate the point `C` in the line `bar(AD)`.

Consider points `p`, `q`, `r`, `s`, and `t` on line `bar(AD)`. construction of triangle SA sum of 2 sides Which of the following property is noted for points `p` and `t` in the line `bar(AD)`?

  • `bar(Ap)+ bar(pB) < bar(AD)` as evident from the figure
  • `bar(At)+ bar(tB) > bar(AD)` as evident from the figure
  • both the above
  • both the above

The answer is "both the above". It is noted that at a single position between the points `A` and `D`, the point `C` is located such that `bar(AC)+bar(CB)=bar(AD)`.

With given "side-angle-sum of two sides" we constructed `bar(AB)`, and `bar(AD)=bar(AC)+bar(CB)` at the given angle `/_A`.
The point `C` is visualized in `bar(AD)` such that `bar(AC)+bar(CB)=bar(AD)`. construction of triangle SA sum of 2 sides Which of the following is an useful observation that will help to locate the point `C` along `bar(AD)`?

  • the line segments `bar(CD)` and `bar(CB)` are equal
  • the triangle `/_\ BCD` is isosceles
  • both the above
  • both the above

The answer is "both the above".

With given "side-angle-sum of two sides" we constructed `bar(AB)`, and `bar(AD)=bar(AC)+bar(CB)` at the given angle `/_A`. The point `C` is visualized in `bar(AD)` such that `bar(AC)+bar(CB)=bar(AD)`.

It is observed that `/_\BCD` is isosceles and only angle `/_D` and base `bar(BD)` are available. The position of point `C` is not yet marked. It is shown in the figure to visualize. construction of triangle SA sum of 2 sides Which of the following property help in identifying the point `C`?

  • two angles of an isosceles triangles are equal. `/_CDB = /_DBC`
  • a perpendicular bisector on the base of isosceles triangle passes through the third vertex
  • both the above
  • both the above

The answer is "both the above".

With given "side-angle-sum of two sides" we constructed `bar(AB)`, and `bar(AD)=bar(AC)+bar(CB)` at the given angle `/_A`.

The point `C` is visualized in `bar(AD)` such that `bar(AC)+bar(CB)=bar(AD)`.

It is observed that `/_\BCD` is isosceles and only angle `/_D` and base `bar(BD)` are available.

The position of point `C` is not yet marked. It is shown in figure to visualize.
construction of triangle SA sum of 2 sides Using the property that
   two angles of an isosceles triangles are equal. `/_CDB = /_DBC`
The angle `/_CDB` can be copied using a compass to mark line `vec(BE)`. The point of intersection of `vec(AD)` and `vec(BE)` is the point `C`.
`/_\ABC` is constructed.

With given "side-angle-sum of two sides" we constructed `bar(AB)`, and `bar(AD)=bar(AC)+bar(CB)` at the given angle `/_A`.

The point `C` is visualized in `bar(AD)` such that `bar(AC)+bar(CB)=bar(AD)`.

It is observed that `/_\BCD` is isosceles and only angle `/_D` and base `bar(BD)` are available.

The position of point `C` is not yet marked. It is shown in figure to visualize.
construction of triangle SA sum of 2 sides Using the property that
   the perpendicular bisector on the base of isosceles triangle passes through the third vertex,
The line `vec(MN)` is drawn as bisector of `bar(DB)`. The point of intersection of `vec(AD)` and `vec(MN)` is the point `C`.
`/_\ABC` is constructed.

Construction of Side-Angle-Sum of 2 Sides triangle : construction of triangle SA sum of 2 sides Visualize the point `C` and the `/_\ BCD` is isosceles triangle.

Locate point `C` by one of the following methods.

 •  copy the angle

 •  draw perpendicular bisector on the base.

                            
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