Let's begin with recalling the Geometric Mean Altitude Theorem.
The altitude drawn to the hypotenuse of a right triangle separates the hypotenuse into two segments. The length of this altitude is the geometric mean between the lengths of these two segments.
We are given that Mike is hanging a string of lights on his barn, and he wants to evaluate its height. We also know he can see that he is 15 feet from the barn and his eye level is 5 ft from the ground. Let x represent the height of the barn above Mike's line of vision. We will name the vertices with consecutive letters.
As we can see, BD is an altitude of a right triangle ABC. Therefore, we can write that the length of BD is the geometric mean between the lengths of AD and DC.
DB=sqrt(AD * DC)
15=sqrt(x* 5)
Next, we will solve the above equation for x using inverse operations.
The length of AD is 45 feet. With this information, we can evaluate the height of the barn. To do this, we will add the lengths of AD and DC.
45ft+ 5ft=50ft
The height of the barn is 50 feet.
