Heights and Distances

Relationship between ratios and height and distances

height and distance

Any triangle is well defined if three sides are given or two sides and one angle is given or two angles and and one side is given. The right angled triangle comes under a triangle with given two angles and one side. By well defined I mean a triangle whose all the sides and all the angles can be known. In a right angled triangle one angle is 90°. If the other angle is α Then the third unknown angle is 180° −(90° + α). If one side is known then all the other sides can be known as the sides are in a particular ratio for a given set of angles.

    In the triangle ABC,
  • if hypotenuse (h) and angle α is given then
    b = h cos α
    p = h sin α

  • if base (b) and angle α is given then
    h = b sec α
    p = b tan α

  • if perpendicular (p) and angle α is given then
    h = p cosec α
    b = p cot α

Angle of elevation and angle of depression

Now if we substitute p for height of an object and b for distance of the object from observation point C. ∠ACB is the angle of observation from the ground to the top of the object. Then we can find height if distance is given and vice-versa. This angle is called the Angle of Elevation. ∠DAC is called the Angle of Depression when the point of observation is A. BC is the line from which angle of elevation is measured and DA is the line from which angle of depression is measured.