PQR is a right angled triangle with hypotenuse PQ while PR = √2 and QR = 1. S is the midpoint of PQ and T is the midpoint of PR. The line segments QT and RS intersect at the point U.
Determine Angle QUR.
Angle TQR = arctan(1/sqrt(2)) ~= 35.26438968275465 deg
Angle PQR = arctan(sqrt(2)) ~= 54.73561031724535 deg
Let x = length of segment SR.
x^2 = 3/4 + 1  sqrt(3)cos(PQR) = 1.75  sqrt(3)/sqrt(3) (law of cosines)
x = sqrt(3/4)
sin(QRS)/(sqrt(3)/2) = sin(PQR)/x (law of sines)
sin(QRS) = (sqrt(2)/sqrt(3)) * (sqrt(3)/2) / x
= 1/(x*sqrt(2)) ~= .816496580927722
Angle QRS ~= 54.735610317245
Angle QUR = 180  angle QRS  angle TQR
~= 180  54.735610317245  35.26438968275465
~= 90 degrees

Posted by Charlie
on 20071031 10:41:27 