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Angle in a square (Posted on 2010-11-16) Difficulty: 3 of 5
Points P and Q are respectively located on the sides AB and BC of square ABCD, such that BP = BQ.

H is the base of the perpendicular from point B to the segment PC.

Determine the measure of DHQ.

No Solution Yet Submitted by K Sengupta    
Rating: 3.0000 (1 votes)

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Solution Solution | Comment 1 of 5

The angle DHQ is 90 degrees.
PROOF:
Let ABCD be a unit square,
   |BP| = |BQ| = b,
and F the base of the perpendicular
from H to side AB.
We can then assign co-ordinate pairs
to the eight point:
   A(1,0)         P(b,0)
   B(0,0)         Q(0,b)
   C(0,1)         H(f,h)
   D(1,1)         F(f,0)
From the similar right triangles
PBC and HFB,
    1     |CB|     |BF|     f
   --- = ------ = ------ = ---
    b     |PB|     |HF|     h
                or
              h = b*f            
From the similar right triangles
PBC and PFH,
    1     |CB|     |HF|      h      b*f
   --- = ------ = ------ = ----- = -----
    b     |PB|     |PF|     b-f     b-f
                or
        b-b^2*f = f         
                        b-h     1-h 
 slope(HQ)*slope(HD) = ----- * ----- 
                        0-f     1-f
                         
                        b-b*f     1-b*f
                     = ------- * -------
                          -f       1-f
                        b-b^2*f
                     = ---------
                           -f
                     = -1
Therefore, HQ and HD are perpendicular.
 

  Posted by Bractals on 2010-11-16 21:02:56
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