Question

How many triangles satisfy the following conditions:
A=50°, b=19 ft, a=9 ft.

Answer 1

if there is only one triangle that will satisfy that one, then there could only be one dimension for each sides of the trianle which will satisfied conditions. its side b could oly be 19 and a could only be 9 So if we prove that the third side (let's name it c) could only have one dimension then there will only be one triangle which will satisfy that. now since A is the angle opposite the side a then by Cosine law,
a^2=c^2+b^2-2bccosA
rearranging this we'll have
c^2-2bccosA=a^2-b^2
completing the square we'll have
c^2-2bcosA+b^2cos^2A=a^2-b^2
(c-bcosA)^2=a^2-b^2
c-bcosA=sqrt(a^2-b^2)
c=bcosA+sqrt(a^2-b^2)
now since A is a constant, cosA will be constant too. b and a are also constant and multiplying, adding, getting the squaroot of the difference of the squares of constants will also yield a constant. therefore the RHD will be a constant so c will in turn be a constant also. Since there could only be one dimensions for each sides of the triangle, then there could only be one triangle whicwl satisfy such conditions.