Question

from a point on the bridge across the river the angle of depression of the bank on the opposite sides of the river are 30 degree and 45 degree respectively if the bridge is at the height of 3 m from the bank find the width of the river

Answer 1

If you join the point to the banks and join the banks to each other, you will have a nice triangle. Let's name the point on the bridge A, one bank as B and the other as C, for triangle ABC. Draw a perpendicular AD from A to base BC, which will be the height 3m.

Draw a line XY parallel to BC but it should touch A. This will be the line of sight.

Angle XAB = 30 degrees (angle of depression)
Angle YAC = 45 degrees (Angle of depression)

Therefore
Angle ABC = 30 degrees (Alternate interior angles)
Angle ACB = 45 degrees (Alternate interior angles)

Tan30 = AD / BD
1/√3 = 3 / BD

BD = 3√3

Tan45 = AD/DC
1 = 3/DC

DC = 3

Now BC is the width of the river (it is the distance betweeen two banks)

BC = BD + DC = 3√3 + 3 = 3(1 + √3)
BC = 3 ( 1 + 1.732) = 3 * 2.732

Therefore, BC = 8.196m

Width of the river = 8.196 metres




Taking √3 as 1.732