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From the top of a tower, the angle of depression to a point on the ground is 45

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Question: From the top of a tower, the angle of depression to a point on the ground is 45 degrees. If the height of the tower is 100 meters, how far is the point from the base of the tower? (2020)

Options:

  1. 100 m
  2. 50 m
  3. 70 m
  4. 80 m

Correct Answer: 100 m

Exam Year: 2020

Solution: 

Using tan(45) = 1, Distance = Height = 100 m.

From the top of a tower, the angle of depression to a point on the ground is 45

Practice Questions

Q1
From the top of a tower, the angle of depression to a point on the ground is 45 degrees. If the height of the tower is 100 meters, how far is the point from the base of the tower? (2020)
  1. 100 m
  2. 50 m
  3. 70 m
  4. 80 m

Questions & Step-by-Step Solutions

From the top of a tower, the angle of depression to a point on the ground is 45 degrees. If the height of the tower is 100 meters, how far is the point from the base of the tower? (2020)
  • Step 1: Understand that the angle of depression is the angle formed between the horizontal line from the top of the tower and the line of sight to the point on the ground.
  • Step 2: Recognize that if the angle of depression is 45 degrees, then the angle of elevation from the point on the ground to the top of the tower is also 45 degrees.
  • Step 3: Recall that in a right triangle, the tangent of an angle is the ratio of the opposite side to the adjacent side. Here, the opposite side is the height of the tower (100 meters) and the adjacent side is the distance from the base of the tower to the point on the ground.
  • Step 4: Use the tangent function: tan(45 degrees) = opposite/adjacent. Since tan(45 degrees) = 1, we can set up the equation: 1 = Height / Distance.
  • Step 5: Substitute the height into the equation: 1 = 100 m / Distance.
  • Step 6: Rearrange the equation to find the Distance: Distance = 100 m.
  • Step 7: Conclude that the point on the ground is 100 meters away from the base of the tower.
  • Angle of Depression – The angle formed by a horizontal line and the line of sight to an object below the horizontal line.
  • Trigonometric Ratios – Using tangent (tan) to relate the angle of depression to the height and distance in a right triangle.
  • Right Triangle Properties – Understanding the relationship between the height of the tower and the distance from the base using the properties of right triangles.
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