Puzzle-5 (XAT-2008)

ABCD is a square. P is the midpoint of AB. The line passing through A and perpendicular to DP intersects the diagonal at Q and BC at R. 

If AB =2 then PR = _______?                                      

(A) 1/2        (B) √3/2      (C) √2         (D) 1 (E) None of these  

Solution follows here:

Solution:

In ∆ADM, as ÐAMD = 90⁰,
ÐADP + ÐDAM = 90⁰ 

                                ------(1)

And also  as ÐDAP = 90⁰, this can be split into

ÐMAP + ÐDAM = 90⁰

                                ------(2)

From (1) and (2), ÐADP = ÐMAP

 ÐDAP = 90⁰ = ÐRBP

By A-A-A similarity, ∆ADP ~ ∆ARB

AB/AP = BR/AP => 1 = BR/1 => BR = 1

Now on right angle ∆PBR, applying Pythagoras theorem,

PR² = PB² + BR² = 1 = 1 = 2

PR = √2

Answer (C)