Non-Right-Angled Triangles (Copy)
1.
In ΔABC, A=40°, a=8 cm, b=10 cm.
Find angle B (to 1 d.p.).
2.
ΔPQR has P=35°, Q=75°, side p=12 cm.
Find side q.
3.
In ΔXYZ, x=7 cm, y=9 cm, ∠Z=120°.
Find z.
4.
In ΔDEF, d=8 cm, e=6 cm, ∠F=50°.
Find f.
5.
Triangle has sides 5 cm, 7 cm, included angle 60°.
Find the third side.
6.
ΔABC has A=65°, a=14 cm, b=16 cm.
Find B.
7.
ΔPQR has p=10 cm, q=12 cm, ∠R=110°.
Find r.
8.
Find area of triangle with sides 7 cm and 9 cm including angle 120°.
9.
Find area of triangle with a=12 cm, b=15 cm, ∠C=40°.
10.
ΔXYZ has ∠X=30°, ∠Y=105°, z=20 cm.
Find x.
11.
ΔABC has A=75°, a=18 cm, b=10 cm.
Find B.
12.
ΔDEF has d=14 cm, e=15 cm, ∠F=100°.
Find f.
13.
ΔPQR has sides p=9 cm, q=11 cm, r=13 cm.
Find angle P.
14.
ΔXYZ has sides x=8, y=10, angle Z=120°.
Find z.
15.
Find area of Δ with sides 10 cm, 12 cm, included angle 50°.
16.
ΔABC has A=40°, a=9 cm, b=11 cm.
Find B.
17.
ΔPQR has p=6 cm, q=8 cm, ∠R=60°.
Find r.
18.
ΔXYZ has ∠X=35°, ∠Y=75°, z=15 cm.
Find y.
19.
ΔABC has sides a=7, b=8, c=9.
Find angle C.
20.
Find area of Δ with a=12, b=14, ∠C=85°.
21.
ΔDEF has D=50°, E=65°, f=20 cm.
Find d.
22.
ΔPQR has p=18, q=20, ∠R=120°.
Find r.
23.
Triangle has sides 13, 14, included angle 100°.
Find third side.
24.
ΔXYZ has x=9, y=10, z=11.
Find ∠Z.
25.
ΔABC has A=80°, B=45°, side a=12 cm.
Find b.
26.
Find area of Δ with sides 8 cm and 10 cm including angle 135°.
27.
ΔPQR has p=15, q=9, r=12.
Find angle R.
28.
ΔXYZ has sides x=7, y=7, z=10.
Find ∠Z.
29.
ΔABC has A=40°, a=6 cm, b=5 cm.
Find possible values of angle B (ambiguous case).
30.
ΔDEF has d=20, e=24, ∠F=90°.
Find f.
1.
sinB/b = sinA/a → sinB/10 = sin40°/8.
sinB = 10×0.6428/8 = 0.8036.
B = 53.1°.
2.
∠R=70°. By sine rule: q/sinQ = p/sinP → q/sin75° = 12/sin35°.
q = 12×0.9659/0.5736 ≈ 20.2 cm.
3.
Cosine rule: z²=7²+9²−2×7×9cos120°.
=49+81−2×63×(−0.5)=130+63=193.
z=√193 ≈ 13.9 cm.
4.
f²=8²+6²−2×8×6cos50°.
=64+36−96×0.6428=100−61.7=38.3.
f=√38.3= 6.2 cm.
5.
c²=5²+7²−2×5×7cos60°.
=25+49−70×0.5=74−35=39.
c=√39= 6.2 cm.
6.
sinB/16=sin65°/14.
sinB=16×0.9063/14=1.036>1 → no solution (ambiguous, triangle not possible).
7.
r²=10²+12²−2×10×12cos110°.
=100+144−240×(−0.342)=244+82.1=326.1.
r=√326.1≈ 18.1 cm.
8.
Area=½ab sinC=½×7×9×sin120°.
=31.5×0.866= 27.3 cm².
9.
Area=½×12×15×sin40°=90×0.6428= 57.9 cm².
10.
∠Z=45°. Use sine rule: x/sinX=z/sinZ.
x/sin30=20/sin45.
x=20×0.5/0.7071=14.14 cm.
So 14.1 cm.
11.
sinB/10=sin75°/18.
sinB=10×0.9659/18=0.5366.
B=32.5°.
12.
f²=14²+15²−2×14×15cos100°.
=196+225−420×(−0.1736)=421+72.1=493.
f=√493= 22.2 cm.
13.
cosP=(q²+r²−p²)/(2qr)=(11²+13²−9²)/(2×11×13).
=(121+169−81)/286=209/286=0.730.
P=43.1°.
14.
z²=8²+10²−2×8×10cos120°.
=64+100−160×(−0.5)=164+80=244.
z=√244= 15.6 cm.
15.
Area=½×10×12×sin50°.
=60×0.766= 46.0 cm².
16.
sinB/11=sin40°/9.
sinB=11×0.6428/9=0.785.
B=51.7°.
17.
r²=6²+8²−2×6×8cos60°.
=36+64−96×0.5=100−48=52.
r=√52= 7.2 cm.
18.
∠Z=70°.
y/sinY=z/sinZ.
y/sin75=15/sin70.
y=15×0.966/0.940=15.4 cm.
19.
cosC=(a²+b²−c²)/(2ab)=(49+64−81)/112=32/112=0.286.
C=73.4°.
20.
Area=½×12×14×sin85°.
=84×0.996= 83.7 cm².
21.
∠F=65°. ∠D=65°.
d/sin50=f/sin65.
d=20×0.766/0.906=16.9 cm.
22.
r²=18²+20²−2×18×20cos120.
=324+400−720×(−0.5)=724+360=1084.
r=√1084= 32.9 cm.
23.
Third²=13²+14²−2×13×14cos100°.
=169+196−364×(−0.1736)=365+63.2=428.
Third=√428= 20.7 cm.
24.
cosZ=(x²+y²−z²)/(2xy)=(81+100−121)/180=60/180=0.333.
Z=70.5°.
25.
∠C=55°.
b/sin45=a/sin80.
b=12×0.707/0.985=8.61 cm.
26.
Area=½×8×10×sin135°.
=40×0.707= 28.3 cm².
27.
cosR=(p²+q²−r²)/(2pq)=(225+81−144)/(270)=162/270=0.600.
R=53.1°.
28.
cosZ=(x²+y²−z²)/(2xy)=(49+49−100)/(98)=−2/98=−0.0204.
Z=91.2°.
29.
sinB/b=sinA/a.
sinB/5=sin40°/6.
sinB=5×0.643/6=0.536.
B=32.5° or 180−32.5=147.5°.
So B=32.5° or 147.5°.
30.
f²=20²+24²−2×20×24cos90°.
=400+576=976.
f=√976= 31.2 cm.