Construction of a Triangle

Construction

The process of drawing geometric figures using only an unmarked ruler and a compass is called construction.
  • The unmarked ruler is used to draw straight lines or extend line segments.
  • The compass is used to draw circles or transfer lengths.
For example, to construct a line segment with the same length as a given line segment , follow these steps:
  1. Draw a straight line using the unmarked ruler, and mark a point on .
  2. Use the compass to measure the length of .
  3. With the compass set to the length of , draw a circle centered at . Let the intersection of the circle and be . Then, .

Opposite Sides and Angles in a Triangle

A triangle is denoted as .
  • The side opposite to is , called the opposite side of .
  • Similarly, is the opposite angle of the side .
    In general, in , the sides opposite to , , and are denoted by , , and , respectively.
Among all segments connecting and , the shortest is . Therefore, in ,
This shows that in a triangle, the length of any side is less than the sum of the other two sides.

Construction of a Triangle with Given Side Lengths

To construct a triangle with sides of length , , and , follow these steps:
  1. Draw a straight line , and mark a segment of length on .
  2. Draw a circle centered at with radius , and another circle centered at with radius . Let the intersection of the circles be .
  3. Connect to and to to complete .

Construction of a Triangle with Two Sides and the Included Angle

To construct a triangle with two sides of length and , and included angle :
  1. Construct an angle equal to .
  2. On and , mark points and such that and , respectively.
  3. Connect and to complete .

Construction of a Triangle with One Side and Two Adjacent Angles

To construct a triangle with one side of length and adjacent angles and :
  1. Draw a straight line , and mark a segment of length on .
  2. Construct angles and equal to and , respectively.
  3. Let the intersection of and be . Connect to and to to complete .