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# Solving Triangles

Trigonometric problem solving culminates in this chapter. Leave no side and no angle unmeasured!

# Solving Triangles: Level 3 Challenges

Find $$PC.$$

Note: This length can be expressed as $$\sqrt{\dfrac{A}{B}}$$, where $$A,B$$ are coprime positive integers. Submit the value of $$A+B$$ as your answer.

Aeroplanes A and B are flying with constant speed in the same vertical plane at angles 30° and 60° with respect to the horizontal, respectively, as shown in the figure.

The airspeed of A is $$100 \sqrt{3}$$ meters per second. Initially, an observer in A sees B directly ahead and at the same altitude at a (horizontal) distance of 500 m.

Assuming they take no evasive action and collide, after how many seconds does this happen?

An ant is lost in a square, and his distances to the vertices of the square are 7, 35, 49 and $$x.$$ Find $$x.$$

• The image is not drawn to scale.

Quadrilateral $$ABCD$$ has $$AB=CD,$$ and the acute angles $$B$$ and $$C$$ satisfy $$\sin B = \frac{4}{9}$$ and $$\sin C= \frac{5}{6}.$$ Find $\frac{\text{area of }\color{red}{\triangle ABC}}{\text{area of }\color{blue}{\triangle BDC}}.$

In triangle $$ABC$$, $$\angle ACB = 90^\circ$$. $$A, D, E,$$ and $$B$$ are consecutive points on $$AB$$ such that $$\overline{AD}=\overline{DE}=\overline{EB}$$. If there exists $$\theta$$ such that $$\overline{CD} = 5 \cos \theta$$ and $$\overline{CE} = 5 \sin \theta$$, what is $$\overline{AB}^2?$$