Distance Survey Problem

In conducting a land survey, the following problem arose. There were two points A and B along a road and points B and D off the road along the respective perpendiculars to the road at A and B. There were buildings on the property that prevented direct measurement of the distances along BD and AC. Measurements, however, could be made for AD, BC, and AB as follows:

AB = 240 yards

AD = 260 yards

BC = 300 yards

A light pole is to be installed at point E, the intersection of CB and AD. How far will the pole be from the road. That is what is the distance EF? Again, an existing building blocks direct measurement.

Method I: Similar triangles

Consider the following diagram

Notice, ∆ACE and ∆DBE are similar triangles since all the angles are congruent. Angles 1 and 4 are vertical angles, and (3 and 6) and (2 and 5) are alternate interior angles with parallel lines CA and DB cut by transversals CB and AD.

What is the ratio between ∆ACE and ∆DBE?

One way to find this ratio is to find the length of AC and DB. The Pythagorean Theorem works nicely here with ∆ABC and ∆ADB.

From this result, it can be determined that ∆ACE is the size of ∆DBE.

Using this ratio, the length of AE and BE can be determined:

Now consider :

The area of this triangle can be found in two ways, using A = ½base*height and Heron's formula:

Heron's Formula:

Using this formula, the area was found to be

Using the formula A = ½base*height, the height (distance from the light pole to the road) can be calculated to be .

Possible Extensions:

Interestingly enough, this result could have been found in a much simpler way if you know a bit about harmonic means. How does this problem relate to harmonic means?