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The Laws of Sines and Cosines · Paper 3 — Free Maths AA HL Practice Questions

1ChallengeLAQLaw of Sines and its applicationss10 marksPaper 3~15 min

In a triangle

ABC

, let

a

b

c

denote the lengths of sides opposite vertices

A

B

C

respectively, and let

R

denote the circumradius. A surveyor measures a triangular plot of land

PQR

. The angle

\angle QPR = 75^\circ

, the side

PQ = 120\,\text{m}

, and the angle

\angle PQR = 60^\circ

(a)

Prove that for any triangle

ABC

\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R

where

R

is the circumradius. [5 marks]

(b)

Using the result from part (a), find the exact values of

PR

and the circumradius

R

of triangle

PQR

. [3 marks]

(c)

A colleague suggests that the surveyor could instead have measured

\angle PQR = 45^\circ

, with

PQ = 120\,\text{m}

and

QR = 60\sqrt{6}\,\text{m}

unchanged. Determine the number of distinct triangles

PQR

satisfying these three measurements, and justify your answer. [2 marks]

Solutions

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