Let's begin with applying the given information to the diagram. We will express all the dimensions only in feet.
We are asked to evaluate the length of segment AE. To do this, notice that, according to the Segment Addition Postulate, we can rewrite the length of AE as a sum of the lengths of AD,DB and BE.
AE=AD+DB+BE
Let's start with evaluating the length of AD using the Geometric Mean Altitude Theorem.
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 segments.
Looking at △ ABC, we can see that, according to this theorem, the length of DC is the geometric mean between the lengths of AD and DB.
DC=sqrt(AD* DB) ⇓
6 412=sqrt(AD* 5)
We will solve the above equations for AD. Our first step will be to rewrite the mixed number on the left-hand side as a fraction. Let x represent the length of AD.
The length of AD is approximately 8 feet. Now, we will find the length of BE. To do this, let's recall the Geometric Mean Leg Theorem.
Geometric Mean Leg Theorem
The altitude drawn to the hypotenuse of a right triangle separates the hypotenuse into two segments. The length of a leg of this triangle is the geometric mean between the length of the hypotenuse and the segment of the hypotenuse adjacent to that leg.
Using this theorem, we can write an equation.
BE=sqrt(BG* BF)
Since we are given that △ BEF is isosceles, the altitude EG divides BF into two congruent segments. This means that the length of BG is the half of the length of BF.
BE=sqrt((1/2*10 1012)*10 1012)
We will solve the above equation. Let y represents the length of BE. Our first step will be to simplify the mixed numbers on the right-hand side.
The length of BE is approximately 7.66 feet. Finally, we will find the approximate length of AE by adding the appropriate side lengths.
AE=AD+DB+BE ⇓
AE≈ 8+ 5+7.66 ⇓
AE≈ 20.66
The length of AE is approximately 20.66 feet.
