Question

How many different triangles of the same area can be drawn without changing the length of two sides?

Answer 1

If we use the formula for area = 1/2 ab sin C and the area and a and b are known, then we can use

sin C = 2 (area) / ab to find angle C as arcsin (2 (area) / ab) to constuct a triangle with that area and the given sides included in the angle. The actual length of the third side will take some figuring, probably with the law of cosines.

I am tempted to try this.

Let’s say the area is 100 square inches and the two sides are 20 inches and 30 inches. Then angle C will be arcsin (200 / 600) = approx 19.47 degrees

Answer 2

just one.. which would be any..

but without changing the length of two sides triangle cannot be changed