triangle solving

Let us consider skew-angled trianglesMathworldPlanetmath.  If one knows three parts of a triangle, among which at least one side, then the other parts may be calculated by using the law of sines and the law of cosines.  We distinguish four cases:

  1. 1.

    ASA.  Known two angles and one side, e.g. α, β, c.  Other parts:

    Figure 1: ASA (angle-side-angle)
  2. 2.

    SSS.  Known all sides a, b, c.  The angles are obtained from

    Figure 2: SSS (side-side-side)
  3. 3.

    SAS.  Known two sides and the angle between them, e.g. b, c, α.  Other parts from

    Figure 3: SAS (side-angle-side)
  4. 4.

    SSA.  Known two sides and the angle of one of them, e.g. a, b, α.  Other parts are gotten from

    Figure 4: SSA (side-side-angle)

    Since the SSA criterion alone does not prove congruencePlanetmathPlanetmath, it is not surprising that there may not always be a single solution for β here. In fact, if the first equation gives  sinβ>1,  then the situation is impossible and the triangle does not exist.  If the equation gives  sinβ<1,  one gets two distinct values of β; an acute β1 and an obtuse  β2=180-β1.  If in this case  β1>α,  then there are two different triangles as , but if  β1α,  then there is only one triangle.

  • source code for the above diagrams

Title triangle solving
Canonical name TriangleSolving
Date of creation 2013-03-22 15:44:02
Last modified on 2013-03-22 15:44:02
Owner stevecheng (10074)
Last modified by stevecheng (10074)
Numerical id 11
Author stevecheng (10074)
Entry type Definition
Classification msc 51-00
Related topic Congruence