Question

Suppose ABCD is a rectangle. Find AB and AD if point M is the midpoint of

BC
,
AM
⊥
MD
, and the perimeter of ABCD is 34 in.

Answer 1

see the attached figure to better understand the problem

we know that
AM=MD
so
triangle AMD is an isosceles right triangle
therefore
its height is half its width.
Then
AB = (1/2)AD----------Equation 1
Perimeter=2*[AB+AD]=34 in ---------> AB+AD=17-------> Equation 2

I substitute 1 in 2
(1/2)AD +AD = 17
(3/2)AD=17
AD=17*2/3----------> AD=34/3-------> 11 1/3 in
AB=(1/2)AD--------> AB=(1/2)*34/3--------> AB=17/3-------> AB=5 2/3 in

the answers are
AD=11 1/3 in
AB = 5 2/3 in