How many unique triangles are there if you are given 4 bits of info and can only use 3 at a time

Question

I have 4 bits of infomation, 2 sides and 2 angles. 2cm, 10cm, 90 degrees and 60 degrees. I can only used 3 at once, so for example: 90 degrees, 2cm and 10cm and I have to arrange them in as many unique triangles as possible and find out the area of the triangle using 1/2 x a x b x Sin(C). I want to find out the number of unique triangles and each of their areas.

Answer 1

I. using two angles and side

use 60, and 90, know other angle is 30.

A. use side= 2cm could be any side, so three triangles possible

B. use side=10 cm could be any side, so three more triangles possible.

II using two sides and 90 degree angle, only two triangles valid (cannot have hypotenuse =2 if one side is 10)

III. using two sides and 60 degree angle, only two triangles are valid (trouble with a=2,b=10, A=60)

10 triangles possible, 8 of them right triangles so area has sin(90)=1

the legs of the right triangles are, (2,√12),(1,√3),(2,2/√3),(10,√300),(5,√75),(10,10/√3),(2,10),(2,√96)

for the triangle a=10, b=2, C=60 degrees --- use formula

for triangle a=10,b=2, A=60 degrees, using law of sines find out C=115 degrees --- use formula

Answer 2

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