 Hello and welcome to the session. In this session we will discuss a question which says that Sam and Mike leave for the club. They travel 11 kilometers from their home at a bearing of 34 degrees and then change direction to a bearing of 115 degrees and travel another 7 kilometers. In that is, how far are Sam and Mike from their home and we thought as at what bearing must Sam and Mike travel to return to their home. Now before starting the solution of this question we should know some results. First is law of science. Let A with C be only triangle with small a, small b and small c representing the measures of sides opposite to angles with measurement A, B and C respectively. Then sine of angle A upon side represented by small a is equal to sine of angle B upon side B is equal to sine of angle C upon side C. Here we can write it as side A upon sine of angle A is equal to side B upon sine of angle B is equal to side C upon sine of angle C. And law of science states that for this triangle the following equations are true. First is A square is equal to B square plus C square minus 2 B C into cos of angle A. Second is B square is equal to A square plus C square minus 2 A C into cos of angle B. And third is C square is equal to A square plus B square minus 2 AB into cos of angle C. Now these results will work out as a key idea for solving out the given question. Now let us start with the solution of the given question. Now in the question we have given that Sam and Mike leave for the class. They travel 11 kilometers from their home at a bearing of 44 degrees and then change direction to a bearing of 115 degrees and travel another 7 kilometers. Now let us make its figure. Now bearing means angle made with the north direction. Now this is the north direction. This is the east direction. This is south and this is west direction. Now since this is north direction and from home they travel at bearing of 44 degrees. So angle made with north direction is 44 degrees and starting point is A and from this point they travel 11 kilometers and reach at this point. At this point we B from here they change direction to a bearing of 115 degrees. So this angle is 115 degrees with north direction and they travel another 7 kilometers to reach club. Let the reach point C after travelling 7 kilometers now join point A and point C. So we have triangle ABC where side AB that is small c is equal to 11, one side BC that is small a is equal to 7. Now we have to find how far they are from their home that is we have to find distance AC which is represented by small b. Now for finding side B we will use this second equation of log cosines. Now we already know measure of side A and measure of side C and now we will have to find measure of angle B. Now let this angle be angle 1 and this be angle 2. Now let us represent this line by L and this line by M. Now this vertical line M is a straight line. So here you can see angle 1 plus 115 degrees that is this angle plus this angle is equal to 180 degrees as they form linear pair angles. So this implies angle 1 is equal to 180 degrees minus 115 degrees which implies angle 1 is equal to 65 degrees. Also you see two vertical lines are parallel that is line L is parallel to line M and AB is transversal. So these two that is angle 2 and 44 degrees that is this angle forms alternate pair angles so they are equal. So we have angle 2 is equal to 44 degrees because they are interior alternate angles. Now this angle B is equal to angle 1 plus angle 2 which implies angle B is equal to 65 degrees plus 44 degrees that is equal to 109 degrees. Now we have got the measurement of sides A and C and also angle B is equal to 109 degrees. Now putting all these values in this equation we have B square is equal to A square that is 7 square plus C square that is 11 square minus 2 into 7 into 11 into cos of B that is cos of 109 degrees which implies B square is equal to 49 plus 121 minus now 2 into 7 into 11 is 154 into cos of 109 degrees is minus 0.32 this implies B square is equal to 170 plus 49.28 which implies B square is equal to 219.28 taking positive square root we get B is approximately equal to 14.8 thus Sam and Mike are approximately 14.8 kilometers away from their home. Now we have to find at what bearing must Sam and Mike travel to return to their home. So in the figure we see that now they will change the direction to follow path AC so that they can reach home. Now we have to find the bearing when they change direction. So here we have this vertical line at point C showing north direction and we have to find this angle. Now let us denote this vertical line by P and this angle by angle 3. Now here we can see vertical lines M and P are parallel to each other and BC is the transversal. So angle 3 is equal to 65 degrees that is this angle because they are interior alternate angles. Now we know that angle around the point is 360 degrees. So this total angle is 360 degrees and this is angle C. So the bearing equal to 360 degrees minus this angle 3 that is 65 degrees minus angle C. Now we have to find angle C. Now we know the measurement of sides B and C. We know the measurement of angle B so using law of signs we have sign of angle B upon side B is equal to sign of angle C upon side C. So let us put the values. So this implies sign of angle B that is sign of 109 degrees upon side B that is 14.8 is equal to sign of angle C upon side C that is 11. So sign of angle C is equal to 11 into sign of 109 degrees upon 14.8 which implies angle C is equal to 11 into Now sign of 109 degrees is equal to 0.95 upon 14.8 and this implies sign of angle C is equal to 0.706. And this gives C that is angle C is equal to sign inverse of 0.706 and this implies angle C is approximately equal to 45 degrees. So bearing is equal to 360 degrees minus angle C that is 45 degrees minus 65 degrees that is equal to 250 degrees. Thus, mic and fan should turn to a bearing of around 250 degrees to return to their home. So this is the solution of the given question. That's all for this session. Hope you all have enjoyed the session.