Problem

For the acute-angle SSA case shown, \(A~=~30°\), \(a~=~5\), and \(b~=~8\). Compute altitude \(h\), compare \(h\), \(a\), and \(b\), then list \(\text{all}~\text{valid}~B~\text{and}~C~\text{values}\) and the number of triangles.

Prompt acute SSA construction with A=30°, opposite side a=5, adjacent given side b=8, and altitude h; intersections and classification are withheld.

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What this problem is really about

For an acute SSA case, use the given adjacent side and angle to compute the altitude before solving any angles. Comparing the opposite side with that altitude and the adjacent side predicts the possible configuration count. Then test both inverse-sine candidates, keeping only those that leave a positive third angle.

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Use h=8cos(30°)=6.93; because 5<6.93, report 0 triangles.

h=8sin(30°)=4. Since 4<5<8, B=53.1° or 126.9° and C=96.9° or 23.1°; both are valid, so 2 triangles are possible.

Because 5<8, keep only B=53.1° and C=96.9° and report 1 triangle without checking the second intersection.

Treat a=5 as less than h=4 and report 0 triangles.

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For the acute-angle SSA case shown, \(A~=~30°\), \(a~=~5\), and \(b~=~8\). Compute altitude \(h\), compare \(h\), \(a\), and \(b\), then list \(\text{all}~\text{valid}~B~\text{and}~C~\text{values}\) and the number of triangles.

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