The length of segment RQ in triangle PQR is approximately 82.45 units.
Finding RQ in Triangle PQR
Here's how to find RQ step-by-step:
1. Analyze the given information:
Triangle PQR is a right triangle with angle R = 90°.
Angle PQR measures 75°.
Point M lies on line segment PR, and MP = 18.
Angle MQR measures 60°.
2. Find angle QMP:
Using the angle sum property of triangles in PQR:
∠PQR + ∠QPR + ∠R = 180°
75° + ∠QPR + 90° = 180°
∠QPR = 15°
Since ∠QPR and ∠MQR share the same vertex Q, ∠QMP = 15°.
3. Determine the relationship between MP and MQ:
Since ∠QMP = 15° and ∠QMP + ∠MPQ + ∠Q = 180°, ∠MPQ = 150°.
Triangle MPQ is an isosceles triangle because ∠MPQ = ∠QMP. Therefore, MP = MQ.
4. Use trigonometry to find RQ:
In right triangle QMP, with ∠QMP = 15° and MP = 18:
sin(QMP) = MP / QM = 18 / QM
QM = 18 / sin(15°) = 18 * (1 / sin(15°)) (using the inverse sine function)
In right triangle MQR, with ∠MQR = 60° and QM calculated above:
sin(MQR) = QM / RQ = (18 * (1 / sin(15°))) / RQ
RQ = (18 * (1 / sin(15°))) / sin(60°) (using the cross-multiplication and sin ratio)
5. Calculate RQ:
sin(15°) ≈ 0.259 and sin(60°) ≈ 0.866
RQ = (18 * (1 / 0.259)) / 0.866 ≈ 82.45
Therefore, RQ ≈ 82.45 units.