 Hello and welcome to the session. In this session we are going to discuss the following question and the question says that Samayra chose to take a walk on a path that makes an angle of 47 degrees with the straight road. After walking 326 meters, she turns 80 degrees to her right and heads back towards the road. How far does Samayra need to walk on her current path to get back on the road? When Samayra returns to the road, how far along the road is she from where she started? We know that law of science states let triangle A, B, C be any triangle with A, B and C representing the measures of sites opposite to angles with measurement A, B and C respectively. Then sin of A upon A is equal to sin of B upon B and is equal to sin of C upon C. With this key idea we shall proceed to the solution. We are given that Samayra starts walking on a path which makes an angle of 47 degrees with a straight road. After walking 326 meters, she turns 80 degrees to her right and heads back towards the road. Let us first draw its figure. Let this be a straight road at an angle of 47 degrees. There is a path. Now Samayra walks 326 meters on this path. Then she turns 80 degrees to her right and starts walking on this path so that she reaches the straight road. Now this will be her current path which meets the road. Now let us label this point as A, this point as B and this point as C. Now we see that these points form a triangle A, B, C where measure of angle A is 47 degrees. Side A, B is equal to 326 meters. Now see that she turned 80 degrees to right so we have this exterior angle as 80 degrees. Now if we see in this figure A, B is a straight line so this total angle is 180 degrees. So we can find measure of angle B in this triangle and it will be 180 degrees minus 80 degrees which is equal to 100 degrees. So we have measure of angle B as 100 degrees. So in triangle A, B, C we have measure of angle A which is equal to 47 degrees. Measure of angle B which is equal to 100 degrees and side A, B which we denote as C is equal to 326. Now since we know angle A at angle B we can find angle C. We know that sum of all angles of a triangle is equal to 180 degrees so in this triangle measure of angle A plus measure of angle B plus measure of angle C should be equal to 180 degrees which implies that measure of angle A that is 47 degrees plus measure of angle B that is 100 degrees plus measure of angle C should be equal to 180 degrees. This implies that measure of angle C should be equal to 180 degrees minus of 47 degrees plus 100 degrees which implies that measure of angle C is equal to 180 degrees minus 147 degrees which further implies that measure of angle C is equal to 33 degrees. Now we have to find how far does Samira need to walk on her current path to get back on the road that is we have to find length of side BC. Let us denote the length of side BC by A and let us denote length of side AB that is 326 meters by C. Since we know measure of angle C measure of angle A and its adjacent side C since AAS that is angle angle side is given So now we use law of signs to solve it. From the key idea we have this equation that is sin of A upon A is equal to sin of C upon C. On putting the values of angle A, angle C and C in this equation we get sin of 47 degrees upon A is equal to sin of 33 degrees upon C that is 326. This can be written as A is equal to 326 into sin of 47 degrees whole upon sin of 33 degrees. This further implies that A is equal to 326 into. Now using calculator we get the value of sin of 47 degrees as 0.73 and the value of sin of 33 degrees as 0.54. So we get A is equal to 326 into 0.73 whole upon 0.54 which implies that A is equal to 440.70 which is approximately equal to 441. So we can say that Samayra walked 441 meters approximately on the current path to reach the road. Now we have to find after reaching the road how far along the road is she from where she started. That is we have to find the length of side AC and let it be denoted by B. Again using law of signs we have sin B upon B is equal to sin C upon C. Now we put values of angle B angle C and C in this equation and we get sin of 100 degrees upon B is equal to sin of 33 degrees upon C that is 326. Now this can be written as B is equal to 326 into sin of 100 degrees whole upon sin of 33 degrees. This is further equal to B is equal to 326 into. Now using calculator we get the value of sin of 100 degrees and value of sin of 33 degrees. Now sin of 100 degrees is 0.98 and sin of 33 degrees is 0.54. So we get the value of B as 326 into 0.98 whole upon 0.54. On solving further we get the value of B as 591.6 approximately. So we say that Samira is approximately 591.6 meters away from where she started. That is this distance is approximately equal to 591.6 meters. This is the required answer. This completes our session. Hope you enjoyed this session.