MCR 3U Grade 11 Un iversity
The Sine Law
The Ambiguous Case
1. Find the measures of the angles x and y, to the nearest tenth of a
degree.
2. Determine the number of possible triangles that could be drawn with
the given measure. Then, find the measures of the other angles in each
possible triangle, Round to the nearest tenth of a degree, if necessary.
a) ∆𝐴𝐵𝐶, where ∠𝐴 = 420 , 𝑎 = 30 cm, and 𝑏 = 25 cm
b) ∆𝐴𝐵𝐶, where ∠𝐵 = 270 , 𝑏 = 25 cm, and 𝑐 = 30 cm
c) ∆𝑃𝑄𝑅, where ∠𝑃 = 300 , 𝑝 = 24 cm, and 𝑞 = 48 cm
d) ∆𝐾𝐿𝑀, where ∠𝑀 = 37.30 , 𝑚 = 85 cm, and 𝑙 = 90 cm
e) ∆𝑈𝑉𝑊, where ∠𝑊 = 38.70 , 𝑤 = 10 cm, and 𝑣 = 25 cm
f) ∆𝐴𝐵𝐶, where ∠𝐵 = 480 , 𝑐 = 15.6 m, and 𝑏 = 12.6 m
g) ∆𝑋𝑌𝑍, where ∠𝑋 = 1200 , 𝑥 = 40 cm, and 𝑧 = 20 cm
h) ∆𝐷𝐸𝐹, where ∠𝐸 = 1440 , 𝑒 = 10.5 m, and 𝑓 = 12.5 m
3. Solve each triangle. Round the angle measures to the nearest tenth of
a degree and the side length to the nearest tenth of a centimetre, if
necessary.
a) ∆𝐴𝐵𝐶, where ∠𝐴 = 450 , 𝑎 = 30 cm, and 𝑏 = 24 cm
b) ∆𝑋𝑌𝑍, where ∠𝑌 = 32.70 , 𝑦 = 54 cm, and 𝑥 = 25 cm
c) ∆𝑃𝑄𝑅, where ∠𝑅 = 40.30 , 𝑟 = 35.2 cm, and 𝑞 = 40.5 cm
d) ∆𝐹𝐺𝐻, where ∠𝐺 = 1050 , 𝑓 = 3.5 cm, and 𝑔 = 6.1 cm
e) ∆𝑅𝑆𝑇, where ∠𝑇 = 50.20 , 𝑠 = 10.5 cm, and 𝑡 = 7.1 cm
f) ∆𝐷𝐸𝐹, where ∠𝐸 = 71.20 , 𝑒 = 29.5 cm, and 𝑓 = 30.3 cm
g) ∆𝐵𝐶𝐷, where ∠𝐶 = 1430 , 𝑑 = 12.5 cm, and 𝑐 = 8.9 cm
h) ∆𝐿𝑀𝑁, where ∠𝐿 = 42.80 , 𝑙 = 15.8 cm, and 𝑛 = 18.5 cm
4. Calculate:
a) the measures of ∠𝐵𝐶𝐷 and ∠𝐵𝐷𝐴, to the nearest tenth of a degree
b) the length of CD, to the nearest tenth of a centimetre
5. Find the length of KL, to the nearest tenth of a metre.
6. A circle with center C and radius 5.5 cm intersects a line segment AB
at points D and E. If ∠CAB = 48.90 and AC = 6.4 cm, what is the length
of cord DE, to the nearest tenth of a centimetre?
7. Determine the length of the cord PQ in each circle, to the nearest
tenth of a metre.
8. Park light A light in a park can illuminate effectively up to a distance
of 100 m. A point on a bike path is 150 m from the light. The sight line
to the light makes an angle of 230 with the bike path. What length of the
bike path, to the nearest metre, is effectively illuminated by the light?
9. Application A triangular garden is enclosed by a fence. A dog is on a
5-m leash tethered to the fence at point P. One corner, B, of the fence is
6.5 m from P and forms a 410 angle, as shown in the diagram.
Determine the total length of fence that the dog can reach, to the nearest
tenth of a metre, if the dog cannot reach side AC.
12. Road work A mechanical digger is being used to dig a ditch for
laying pipes under a road. The two sections of the arm have lengths of
3.8 m and 2.7 m, as shown. If the operator keeps the first section fixed at
430 to the horizontal, and then moves the second section to scoop out
earth, what length of ditch will be worked in one awing of the second
section, to the nearest tenth of a metre?