sam observed that a hall is on the east side & also noticed that a mosque is in the south side.he walks 100km north to a mall and sees that the hall is bearing 148degre & the mosque is 168degree

Question

Calculate the distance between the starting point from the hall.find the distance between mall and the hall.find the bearing of the mall from the mosque &from Sam starting point.

Answer 1

I hope I have interpreted this problem correctly.

Before Sam goes to the mall (S) he is at point O, the hall (H) is presumably due east and the mosque (M) is south, but not due south.

When Sam reaches the mall after a very long walk (!) the hall's bearing has changed from 090 to 148.

The triangle OSH is right-angled, with the right angle at O and OS=100km.

Angle OSH=180-148=32 degrees; angle MSH is 168-148=20 degrees.

OH/OS=tan32=0.6249 approx. so OH=62.49km, the distance from the starting-point to the hall.

OS/SH=cos32, so SH=OS/cos32=100/0.8480=117.92km, the distance between the mall and the hall.

Angle OSM=180-168=12 degrees, and OSM=SMH, so the bearing of the mall from the mosque is -12 or 360-12=348.

Sine rule: HM/sin20=HS/sin12,

so HM=HSsin20/sin12=OS/cos32 * sin20/sin12=100/0.8480*0.3420/0.2079=193.98km.

tanMOH=HM/OH=193.98/62.49=3.1041 approx., so MOH=72.14 degrees, making the bearing of the mosque from O, the starting point 90+72.14=162.14. 

The bearing of the mall from the starting-point is 0.