The Law of Sines: Practice

The angles $A$ and $B$ have different sines. What expression equals the common side-over-sine ratio in triangle $ABC$?

In an acute SSA construction, the perpendicular foot $H$ is 15 units ahead of $A$. Its distance to the circle center $C$ is 8 units. The circle radius is 10 units. What lengths can $AB$ have?

Two triangles share the same acute SSA data and the same segment $AC$. One uses the circle crossing nearer to $A$ on ray $AB$. Which triangle has greater area?

On one shore, stations $A$ and $B$ are 135 m apart. The interior angles at $A$ and $B$ toward marker $C$ are $27^\circ$ and $83^\circ$. All three points lie in one horizontal plane. How far is $C$ from $A$ to the nearest tenth of a meter?

The known angle is $105^\circ$ and $b$ is 12 units. Which condition on the radius $a$ is necessary and sufficient for a valid triangle?

For a triangle with a fixed known pair $A,a$, which expression gives its perimeter once $B$ and $C$ are known?

In an SSA problem, an inverse sine calculation gives an acute candidate $B_1$. Under exactly which condition does its supplement give a valid triangle?

When angle $A$ measures $45^\circ$ and side $b$ is 26 units long, there is one SSA triangle and it has a right angle at $B$. What must $a$ be?

Angle $A$ of triangle $ABC$ measures $25^\circ$. The lengths of $a$ and $b$ are 14 cm and 24 cm, respectively. What are all the possible lengths of $AB$, to the nearest tenth of a centimeter?

A crew measures angle $A$ as $29^\circ$. The lengths of $a$ and $b$ are 18 m and 24 m, respectively. Two positions on ray $AB$ satisfy these readings. A field note says the angle at $B$ is acute. Which position matches?

Angle $A$ measures $150^\circ$. The lengths of $b$ and $a$ are 20 and 12, respectively. The perpendicular distance from $C$ to the full ray line is 10. Where are the two circle intersections with that full line?

In triangle $ABC$, angles $A$ and $B$ measure $39^\circ$ and $84^\circ$, respectively. Side $c$ is 22 cm long. Which report of $a$ and $b$, respectively, matches the givens to the nearest tenth of a centimeter?