Problem

In the \(SSA\) triangle shown, \(a~=~8\), \(A~=~30°\), and \(b~=~10\). Use the \(\text{Law}~\text{of}~\text{Sines}\) to find every valid value of \(B\), check each resulting angle sum, and state the number of possible triangles.

An SSA ray, fixed base, swing circle, altitude, and unlabeled geometric intersections for evaluating inverse-sine candidates.

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What this problem is really about

An SSA setup needs an ambiguity audit after inverse sine. Compute the principal angle, form its supplement, and test each candidate with the given angle; keep a candidate only if the remaining third angle is positive. The number of surviving angle sets is the number of possible triangles.

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sin(B)=0.625 gives only B=38.7 degrees and C=111.3 degrees, so report 1 triangle without testing the supplementary angle.

Find B=38.7 degrees and 141.3 degrees, but discard 141.3 degrees solely because it is obtuse, leaving 1 triangle.

Use only the supplementary candidate B=141.3 degrees, which gives C=8.7 degrees, and report 1 triangle.

sin(B)=10sin(30 degrees)/8=0.625, so B=38.7 degrees or 141.3 degrees. These give C=111.3 degrees or 8.7 degrees, so both candidates are valid and 2 triangles are possible.

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In the \(SSA\) triangle shown, \(a~=~8\), \(A~=~30°\), and \(b~=~10\). Use the \(\text{Law}~\text{of}~\text{Sines}\) to find every valid value of \(B\), check each resulting angle sum, and state the number of possible triangles.

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