Geometry

Solving Triangles

Solving Triangles: Level 3 Challenges

         

Find PC.PC.

Note: This length can be expressed as AB\sqrt{\dfrac{A}{B}}, where A,BA,B are coprime positive integers. Submit the value of A+BA+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 1003100 \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.x. Find x.x.

Note: The image is not drawn to scale.

Quadrilateral ABCDABCD has AB=CD,AB=CD, and the acute angles BB and CC satisfy sinB=49\sin B = \frac{4}{9} and sinC=56.\sin C= \frac{5}{6}. Find area of ABCarea of BDC.\frac{\text{area of }\color{red}{\triangle ABC}}{\text{area of }\color{blue}{\triangle BDC}}.

Give your answer to 3 decimal places.

  • The image is not drawn to scale.

In triangle ABC ABC, ACB=90\angle ACB = 90^\circ . Points A,D,E,A, D, E, and BB are consecutive points on side ABAB such that AD=DE=EB\overline{AD}=\overline{DE}=\overline{EB}. If there exists θ \theta such that CD=5cosθ \overline{CD} = 5 \cos \theta and CE=5sinθ\overline{CE} = 5 \sin \theta , what is AB2?\overline{AB}^2?

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