Solving Triangles

Concept Quizzes

Triangle Solving Warmup
Lengths in Right Triangles
Sine Rule
Sine Rule - Ambiguous Case
Cosine Rule
Extended Sine Rule
Bearing Problems
Solving Triangles - Problem Solving

Challenge Quizzes

Solving Triangles: Level 2 Challenges
Solving Triangles: Level 3 Challenges
Solving Triangles: Level 4 Challenges

Sine Rule

In triangle $ABC$ above, $\angle BAC =x= 30^{\circ}, \sin \angle ABC = \frac{1}{4}, \lvert \overline{BC} \rvert =a= 9,$ where $\lvert \overline{BC} \rvert$ is the length of $\overline{BC}.$ What is the value of $\lvert \overline{AC} \rvert^2?$

\frac{161}{19}

\frac{81}{4}

\frac{323}{16}

\frac{325}{16}

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by Brilliant Staff

Triangle $ABC$ has angles $\angle A = 60^{\circ}$ and $\angle B = 45^{\circ}$ , and a side length $a = 21 \sqrt{6}$ . What is the side length $b$ ?

Details and assumptions

$a$ and $b$ are the lengths of the sides opposite to the vertices $A$ and $B$ , respectively.

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by Brilliant Staff

In triangle $ABC$ above, we are given the side lengths as follows: $\lvert\overline{BC}\rvert=a, \lvert\overline{CA}\rvert=b, \lvert\overline{AB}\rvert=c.$ If $ab : bc : ca = 2 : 11 : 5,$ what is the value of $\frac{\sin A + \sin B}{\sin C}?$

\frac{65}{4}

\frac{65}{22}

\frac{77}{10}

\frac{32}{55}

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by Brilliant Staff

In triangle $ABC$ above, $\angle ABC =x= 30^{\circ}$ and $\lvert \overline{BC} \rvert =a= 6, \lvert \overline{CA} \rvert =b= 10.$ What is the value of $\left(\sin \angle BAC\right)^2?$

\frac{19}{200}

\frac{9}{100}

\frac{36}{401}

\frac{6}{65}

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by Brilliant Staff

In triangle $ABC$ above, we are given the side lengths as follows: $\lvert\overline{BC}\rvert=a, \lvert\overline{CA}\rvert=b, \lvert\overline{AB}\rvert=c.$ If $\sin A : \sin B = 5 : 4$ and $\ a : c = 2 : 5,$ what is $a : b : c?$

Note: The above diagram is not drawn to scale.

8 : 25 : 10 10 : 8 : 25 25 : 8 : 10 25 : 10 : 8

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by Brilliant Staff