Question

Given rectangle ABCD, segment AC is a diagonal, segment BE is perpendicular to segment AC at E. Segment AE = 8 inches and segment CE = 5 inches. What is the length of segment BE?

Answer 1

Check the picture attached.

Let m(BAE)=m(ACD)=α

(BAE and ACD are congruent, since they are alternate interior angles, or Z angles)

Let m(ABE)=β. 

So in triangle ABE, the measures of the angles are 90, α and β degrees.

This means that m(BCE)=β, since the 2 other angles of triangle BCE are 90 and α degrees.

thus, we have the similarity of triangles ABE and BCE,

so the following rations are equal:

$\frac{AB}{BC} = \frac{BE}{CE} = \frac{AE}{BE}$

so


$\frac{AB}{BC} = \frac{x}{5} = \frac{8}{x}$

so 

$\frac{x}{5} = \frac{8}{x}$

$x^{2} =40$

$x= \sqrt{40}= \sqrt{4*10}=2 \sqrt{10}$   (inches)


Remark, we can also apply Euclid's theorem directly.