The measures of two sides of a triangle and the angle opposite one of them are given. Determine the number of triangles that satisfy the given set of conditions. Please show your solutions.

Question

a = 40.2, b = 52.4, α = 41.5°

Answer 1

Create a diagram of a potential triangle (or triangles) with a horizontal side (length unknown), the given angle as the bottom left corner of the triangle, and extend the given side that is NOT opposite the given angle (in the above question, this side is length b) up from that vertex (and to the right if the given angle is acute, which it usually is for these "ambiguous case" questions).

SOHCAHTOA trig can show us that the perpendicular distance from the endpoint of that extended side back down to the horizontal side is bsinα (in this case 52.4sin41.5° ~ 34.7). This is the value we need to compare the other given side to (in this case, side a).

If a < bsinα , no triangles are possible to construct with these measures (the side opposite the given angle isn't long enough to reach the horizontal side).

If a = bsinα , it is possible to construct exactly 1 (right) triangle with these measures.

If bsinα < a < b , it is possible to construct 2 triangles with these measures, 1 acute, 1 obtuse. This is known as the "ambiguous case." The angles β and β^' (opposite side b) will be supplements.

If b < a. , it is possible to construct exactly 1 triangle with these measures.