 Hi and welcome to the session. I am Shashi and I am going to help you with the following question. Question says, from the top of a 7 meter high building, the angle of elevation of the top of a cable tower is 60 degrees and the angle of depression of its foot is 45 degrees. Determine the height of the tower. First of all, let us understand what is angle of depression. Angle of depression of an object viewed is the angle found by the line of sight with the horizontal when it is below the horizontal level. Here pq is the horizontal line and qr is the line of sight. So angle pqr is the angle of depression here. Clearly we can see angle of depression of r from q is equal to angle pqr. Now this angle is equal to angle of elevation of q from r. Clearly we can see these two lines are parallel lines and this is a transversal. So this angle is equal to this angle as these two angles are alternate interior angles. So we can say angle of depression pqr is equal to angle of elevation of q from r that is angle qrs. Now this is the key idea to solve the given question. Let us now start the solution. First of all let us draw a simple diagram to represent the problem. Now in this diagram let DE is the building. We know height of the building is equal to 7 meters. This is given in the question. Now we know from top of 7 meter high building angle of elevation of top of the table tower is 60 degrees. Let AC be the cable tower and angle of elevation of A from E is equal to 60 degrees. Clearly we can see this is the horizontal line and this is line of sight. Line of sight is above the horizontal line. So angle of elevation is AEB. So we can write let DE is equal to 7 meters be the building AC is the cable tower. Angle AEB is equal to 60 degrees. This is the angle of elevation of the top of our cable tower from the top of our building. Now we are also given that angle of depression of foot of the tower from top of the building is 45 degrees. Now clearly we can see this is the line of sight and this is horizontal line. Line of sight is below the horizontal line. So angle of depression is angle BEC. Angle BEC is equal to 45 degrees. Now from key idea we know angle of depression of C from E is equal to angle of elevation of E from C that is angle ECD is equal to 45 degrees. Clearly we can see BE is parallel to CD and CE is transversal. So angle BEC is equal to angle ECD as their alternate angles. So we can write BE is parallel to CD and CE is transversal. So angle BEC is equal to angle ECD is equal to 45 degrees. Now we have to determine height of the tower. Now we know height of the tower is AC. Here we have drawn horizontal line through E. Horizontal line through E intersect AC at BE. Now we know ED is equal to BC is equal to 7 meters. We know BE is parallel to CD and BC is parallel to ED. So this implies BCDE is a parallelogram and opposite size of parallelogram are equal. So ED is equal to BC. So we can write BC is equal to ED is equal to 7 meters. Now first of all let us consider triangle CED. So we can write in right triangle CED ED upon DC is equal to tan 45 degrees. We know tan theta is equal to perpendicular upon base in this triangle. ED is the perpendicular and CD is the base. Now substituting corresponding values for tan 45 degrees and ED in this expression we get 1 is equal to 7 upon CD. We know ED is equal to 7 meters and tan 45 degrees is equal to 1. Now multiplying both the sides of this expression why CD we get CD is equal to 7 meters. Now we know CD EB is a parallelogram. So CD is equal to BE since opposite size of parallelogram are equal. So we can write CD is equal to BE is equal to 7 meters. Now let us consider right triangle ABE in right triangle ABE. BE is equal to 7 meters and tan 60 degrees is equal to AB upon BE. We know tan theta is equal to perpendicular upon base in this triangle AB is the perpendicular and BE is the base. Now substituting corresponding values of tan 60 degrees and BE in this expression we get root 3 is equal to AB upon 7. We know BE is equal to 7 meters and tan 60 degrees is equal to root 3. Now multiplying both the sides of this expression by 7 we get 7 root 3 is equal to AB or we can simply write it as AB is equal to 7 root 3 meters. Now we have to find height of the tower we know height of the tower is AC and AC is equal to AB plus BC. So we can write AC is equal to AB plus BC. Now this implies AC is equal to we know AB is equal to 7 root 3 meters. So substituting for AB 7 root 3 and here BC we know is equal to 7 meters. So we will substitute 7 for BC. So AC is equal to 7 root 3 plus 7 meters. Now taking 7 common on right hand side we get AC is equal to 7 multiplied by root 3 plus 1 meters. So we get height of the tower is equal to 7 multiplied by root 3 plus 1 meters. So this is our required answer. This completes the session. Hope you understood the solution. Take care and have a nice day.