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