Asked by sl
given: triangle PQR, m<R=90 degrees, MP=18, m<PQR=75 degrees, m<MQR=60 degrees. Find RQ
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GPT-5 mini
AI
RQ = 9.
Reason: Put R at the origin, RQ along the x-axis, so Q = (x,0) with x = RQ, and P = (0, x tan75°). The ray from Q making 60° with QR (inside the triangle) has direction 120° from the +x-axis, so it meets PR (the y-axis) at M = (0, x tan60°) = (0, x√3). Thus
MP = P_y - M_y = x(tan75° - tan60°).
But tan75° = 2 + √3 and tan60° = √3, so tan75° - tan60° = 2. Hence MP = 2x = 18, so x = 9. Therefore RQ = 9.
Reason: Put R at the origin, RQ along the x-axis, so Q = (x,0) with x = RQ, and P = (0, x tan75°). The ray from Q making 60° with QR (inside the triangle) has direction 120° from the +x-axis, so it meets PR (the y-axis) at M = (0, x tan60°) = (0, x√3). Thus
MP = P_y - M_y = x(tan75° - tan60°).
But tan75° = 2 + √3 and tan60° = √3, so tan75° - tan60° = 2. Hence MP = 2x = 18, so x = 9. Therefore RQ = 9.
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