trigonometry math

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A triangular plot has a frontage on the sea of 100 yards. The boundary lines running from the beach make angles of 60 degrees and 50 degrees respectively on the inner side of the lot in line with the shore line. Determine the distance of the dividing line from the vertex of the triangle to the opposite side along the shore line to divide the lots into equal parts.

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To find the distance from the vertex to the shoreline which will divide the lot into two equal parts, find the  length of what would be the line dividing the lot.

As we have 2 exisitng angles, find the third:

All angles =180 degrees. Therefore third angle = 180 - (60 + 50) = 70 degrees.  (Angles of a triangle)

Use the Sine Rule to find the sides opposite the 50 degree or the 60 degree angles. I have used 50 degrees and called the side a and the 60 degree angle and called it side c: 

Therefore, a/sin50 = 100/sin70 (Sine Rule) (Care to use the sides opposite the angles for the ratio)

Therefore, a=81.5 yards (rounded off) and 

c/sin 60 = 100/sin 70 (Sine Rule) = 92.16

In the answer, you need only use one or other of these. 

We know that two equal lots have been created so the shore line has been halved and the length of the shoreline (the base) of the two new equal lots is therefore 50 yds. 

Now we have enough information to find the required distance (ie the length) using the Cosine Rule as we know that both triangles have half of the base (half of 100 yards)(divided into equal parts)

Therefore, Side (squared) = 50^2 + 81.52^2 - 2(50 x 81.52)Cos 60 (Cosine Rule)

Therefore  side ^2 = 5069.51

therefore = 71.2 yards


The distance from the vertex to the shoreline side is 71.2 yards, dividing the lot into equal parts. 

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