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?

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:



so



so 





  (inches)

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