Find the measure of an angleLet ABC a triangle with the measure of angle B equal to 56 degrees and the measure of angle C equal to 36 degrees. P is a point on the side AB, and Q on the side AC so B P = C Q = 1. Let M be the middle of the side BC and B M = M C = 5. Also, be N the middle of the PQ side. Find out the BMN angle measurement.Given that there are given lengths of sides, I try to use the sine theorem in the triangles BPM, respectively CQM. Then I used the formula sin ⁡ ( a ) + sin ⁡ ( b ) = 2 sin ⁡ ( ( a + b ) / 2 ) cos ⁡ ( ( a − b ) / 2 ) hoping to get some measure of that angle. Can you give me a hint, please?

Question

Find the measure of an angleLet ABC a triangle with the measure of angle B equal to 56 degrees and the measure of angle C equal to 36 degrees. P is a point on the side AB, and Q on the side AC so  B  P  =  C  Q  =  1. Let M be the middle of the side BC and   B  M  =  M  C  =  5. Also, be N the middle of the PQ side. Find out the BMN angle measurement.Given that there are given lengths of sides, I try to use the sine theorem in the triangles BPM, respectively CQM. Then I used the formula   sin  ⁡  (  a  )  +  sin  ⁡  (  b  )  =  2  sin  ⁡  (  (  a  +  b  )      /    2  )  cos  ⁡  (  (  a  −  b  )      /    2  ) hoping to get some measure of that angle. Can you give me a hint, please?

antidootnw · Accepted Answer

Step 1You can make the following.By the law of sines get AB and AQ, which gives AP and AQ and by the law of cosines get PQ and PN.Now, by law of cosines get   ∡  A  P  Q and from this get   ∡  B  P  N  ..Step 2Also, by law of cosines get BN, by law of cosines again get   ∡  B  P  N, which gives   ∡  N  B  M  ..Now, in   Δ  N  B  M we have now: BM, BN and   ∡  N  B  M  ..

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