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:


`(AB)/(BC) = (BE)/(CE) = (AE)/(BE)`

so



`(AB)/(BC) = (x)/(5) = (8)/(x)`

so


`(x)/(5) = (8)/(x)`


`x^(2) =40`


`x= √(40)= √(4*10)=2 √(10)` (inches)


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