The value of RQ = 9, from the given details, using the sine rule and other properties of triangles.
What is the sine rule?
According to the sine rule, in a triangle ABC,
a/sin A = b/sin B = c/sin C
where a = BC, b = AC, and c = AB.
How to solve the question?
In the question, we are given a diagram and the following details:
∆PQR, m ∠R = 90° m ∠PQR = 75°, m ∠MQR = 60° M ∈ PR , MP = 18. We are asked to find the value of RQ.
Now, since m ∠PQR = 75° and m ∠MQR = 60°,
we can say that m ∠PQM = m ∠PQR - m ∠ MQR = 75° - 60° = 15°.
In Δ MQR,
m ∠R = 90°, m ∠Q = 60°.
∴ m ∠M = 180° - (m ∠R + m ∠Q) = 180° - (90° + 60°) = 30°. {using angle sum property of triangle}
At point M, m ∠RMQ + m ∠ QMP = 180° {Adjacent angles}
or, 30° + m ∠ QMP = 180° {∵ m ∠RMQ = 30°, calculated above}
or, m ∠ QMP = 180° - 30° = 150°.
Now, in ΔPQM,
m ∠P + m ∠Q + m ∠M = 180°.
or, m ∠P + 15° + 150° = 180°
or, m ∠P + 165° = 180°,
or, m ∠P = 15°.
Since, m ∠P = m ∠Q, Δ QPM is an isosceles triangle, with MP = MQ = 18 {Since, MP = 18 is given}.
Now in ΔRQM, by applying the sine rule, we can say that:
MQ/sin R = RQ/sin M,
or, 18/sin 90° = RQ/sin 30°,
or, RQ = (18*sin 30°)/sin 90° = (18*0.5)/1 {Since, sin 30° = 0.5 and sin 90° = 1},
or, RQ = 9.
Therefore, RQ = 9, using the sine rule.
Learn more about sine rule at
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