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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, Negative difference between 2 Sides

In this page, a short and to-the-point overview of constructing triangles with angle-side-negative difference between two sides is provided. Difference between 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 `/_\ ABC` with given "side-angle-difference between other two sides" `bar(AB)=4`cm, `/_A=50^@`, and `bar(AC)-bar(BC) = -1`cm OR `bar(BC)-bar(AC) = 1`cm.construction of triangle SA difference between 2 sides  •  Construct a line and mark `bar(AB)` measuring `4`cm

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

 •  measure `bar(BC)-bar(AC)` `=1`cm and draw an arc cutting the ray at `D` at the back-end of the ray, as shown in the figure.

This is directly constructed from the given parameters.

With given "side-angle-difference between two sides" we constructed `bar(AB)`, and `bar(AD)=bar(BC)-bar(AC)` at the given angle `/_A`.

The objective now is to locate the point `C` in the line `vec(AD)`.

Consider points `p`, `q`, `r`, `s`, and `t` on line `vec(AD)`. construction of triangle SA difference between 2 sides Which of the following property is noted for points `p` and `t` in the line `vec(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 in `vec(AD)`, the point `C` is located such that `bar(AC)-bar(CB)=bar(AD)`.

With given "side-angle-difference between two sides" we constructed `bar(AB)`, and `bar(AD)=bar(BC)-bar(AC)` at the given angle `/_A`.

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

The point `C` is visualized in `vec(AD)` such that `bar(BC)-bar(AC)=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 difference between 2 sides Which of the following property helps 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-difference between two sides" we constructed `bar(AB)`, and `bar(AD)=bar(BC)-bar(AC)` at the given angle `/_A`.

The point `C` is visualized in `vec(AD)` such that `bar(BC)-bar(AC)=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 to be marked. It is shown in figure to visualize.
construction of triangle SA difference between 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 rays `vec(AD)` and `vec(BE)` is the point `C`.
`/_\ABC` is constructed.

With given "side-angle-difference between two sides" we constructed `bar(AB)`, and `bar(AD)=bar(BC)-bar(AC)` at the given angle `/_A`.

The point `C` is visualized in `vec(AD)` such that `bar(BC)-bar(AC)=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 to be marked. It is shown in figure to visualize.
construction of triangle SA difference between 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 constructed as the 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-Negative difference between 2 Sides of triangle : construction of triangle SA difference between 2 sides Visualize that at point `C`, the `/_\ BCD` is isosceles triangle.
Locate point `C` with one of the following methods.

 •  copy the angle

 •  draw perpendicular bisector on the base.

                            
slide-show version coming soon