How Can Ken use Similar Triangles to find the height of a Building?

Question

Ken is 6 Feet Tall and his Shadow is 10 Feet long. The shadow of the Building is 25 Feet Long.

Savannah Aguilar · Accepted Answer

The height of the building is #=AB=15# ft.

.

Triangles #DeltaABC# and #DeltaDEC# are similar trianglesbecause all three of their corresponding angles are congruent.

Therefore, the ratios of their corresponding sides are equal.

#(AB)/(DE)=(BC)/(EC)#

#(AB)/6=(BC)/10#

Let's solve for #BC#:

#BC=10/6(AB)=5/3(AB)#

#BC=25# ft. (shadow of the building)

Let's plug it in:

#25=5/3(AB)#

#AB=25/(5/3)=25*3/5=15# ft.

Evelyn Arboleda · Answer

undefined Ken can use similar triangles to find the height of a building by following these steps:

1. Stand a known distance away from the building. This distance should be a point where Ken can see the top of the building clearly and measure the angle of elevation to the top of the building.

2. Measure the angle of elevation ($ \theta $) from the observer's eye level to the top of the building. This angle is the same as the angle formed between the observer's line of sight and the horizontal ground.

3. Measure the horizontal distance ($ d $) from the observer to the base of the building. This distance should be measured perpendicular to the building.

4. Set up a proportion using the similar triangles formed by the observer, the building, and a vertical line extended from the top of the building to the ground. The height of the building ($ h $) is to the distance from the observer to the building ($ d $) as the opposite side of the angle of elevation ($ h $) is to the adjacent side ($ d $).

$$ \tan(\theta) = \frac{h}{d} $$

5. Solve the proportion for $ h $, the height of the building:

$$ h = d \times \tan(\theta) $$

Using this method, Ken can determine the height of the building by measuring the angle of elevation and the horizontal distance from his position to the building.