ABCD IS A SQUARE OF SIDE LENGTH 2009. M AND N ARE N POI...

8. ABCD is a square of side length 2009. M and N are

N

points on the extension of the diagonal AC such that

∠MBN=135°. Determine the minimum length of MN.

D

C

【Solution】

Since

∠ = ° − ∠ = ° − ° − ° − ∠ = ∠

AMB ABM CBN CBN

45 45 (135 90 )

A B

.

M

AM

Hence triangles ABM and CBN are similar, so that ²⁰⁰⁹

CN = .

2009

Let CN=x, then

= + +

MN MA AC CN

2

2009 2009 2

x x

( 2009 ) 2 2009 2009 2

= − + × +

≥ × +

2009 (2 2)

ANS: 2009(2 + 2)