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Right Triangle And Midpoints (Posted on 2007-10-31) Difficulty: 3 of 5
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.

See The Solution Submitted by K Sengupta    
Rating: 3.0000 (2 votes)

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Solution solution | Comment 2 of 3 |

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 2007-10-31 10:41:27
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