Only one triangle can be constructed with two side lengths of 10 centimeters and 8 centimeters and an angle of 40° between the sides.
How to determine the number of possible triangles with two known adjacent side length and a common angle
A criteria to determine the number of possible triangles is using the law of cosine, in which the length of the missing side ([tex]c[/tex]) is determined as a function of the two known adjacent sides ([tex]a, b[/tex]) and a common angle ([tex]\theta[/tex]):
[tex]c = \sqrt{a^{2}+b^{2}-2\cdot a\cdot b\cdot \cos \theta}[/tex] (1)
If we know that [tex]a = 10\,cm[/tex], [tex]b = 8\,cm[/tex] and [tex]\theta = 40^{\circ}[/tex], then the length of the missing side is:
[tex]c = \sqrt{(10\,cm)^{2}+(8\,cm)^{2}-2\cdot (10\,cm)\cdot (8\,cm)\cdot \cos 40^{\circ}}[/tex]
[tex]c \approx 6.437\,cm[/tex]
Since there is an unique solution, we conclude that only one triangle can be constructed. [tex]\blacksquare[/tex]
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