PQRS is a convex quadrilateral with diagonals PR and QS that intersect at the point T. It is known that, Angle QPR = 50o, Angle RPS = 60o, Angle RQS = 30o and Angle QSR = 25o
Determine Angle PTQ.
We can make a table based on assumed values of angle PSQ, and taking the length of side SP to be one unit. Call angle PSQ as having x degrees.
Diagonal SQ, in triangle SQR is found via the law of sines: SQ/sin(180-30-25) = sin(30)/1.
Side PS can be found via the law of sines in triangle PRS, where angle PRS = 180 - 60 - 25 - x.
Side PQ can be found via the law of cosines from triangle PSQ and then angle PQS by the law of sines from that triangle. This allows the remaining angle, QPS to be found.
Angle QPR is 60 less than angle QPS, and angle PTQ then is 180 - QPR - PQS.
Assumed PSQ Resulting QPR PTQ
10.00000000000 88.43748013915 70.00000000000
15.00000000000 76.40339332155 75.00000000000
20.00000000000 67.12410354148 80.00000000000
25.00000000000 60.00000000000 85.00000000000
30.00000000000 54.44146394259 90.00000000000
35.00000000000 50.00000000000 95.00000000000
40.00000000000 46.35991910939 100.00000000000
45.00000000000 43.30265221763 105.00000000000
50.00000000000 40.67565190561 110.00000000000
55.00000000000 38.37046283484 115.00000000000
60.00000000000 36.30816333454 120.00000000000
65.00000000000 34.42981391945 125.00000000000
70.00000000000 32.69013629600 130.00000000000
75.00000000000 31.05324821680 135.00000000000
80.00000000000 29.48970527960 140.00000000000
85.00000000000 27.97437254131 145.00000000000
90.00000000000 26.48481555031 150.00000000000
95.00000000000 25.00000000000 155.00000000000
100.00000000000 23.49914579819 160.00000000000
105.00000000000 21.96060874231 165.00000000000
110.00000000000 20.36066781447 170.00000000000
As we are told that QPR should be 50 degrees, angle PTQ is 95 degrees.
DEFDBL A-Z
pi = ATN(1) * 4
sr = 1
sq = SIN(125 * pi / 180) * 2
'start w/ psq = 30
psq = 10
DO
ps = SIN((180 - 60 - 25 - psq) * pi / 180) / SIN(60 * pi / 180)
pq = SQR(ps * ps + sq * sq - 2 * ps * sq * COS(psq * pi / 180))
sinpqs = ps * SIN(psq * pi / 180) / pq
pqs = ATN(sinpqs / SQR(1 - sinpqs * sinpqs)) * 180 / pi
qps = 180 - psq - pqs
qpr = qps - 60
ptq = 180 - qpr - pqs
PRINT USING " ###.###########"; psq; qpr; ptq
psq = psq + 5
LOOP UNTIL psq > 110
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Posted by Charlie
on 2007-10-03 11:22:50 |
solution
