WISKUNDE
GRAAD 11
NOG OEFENINGE
Oplos van driehoeke.
MATHEMATICS
GRADE 11
MORE EXERCISES
Solving triangles.
Vraag / Question 1
Los die volgende driehoeke op (d.i. bepaal
Solve the following triangles (that is determine
die grootte van elke onbekende sy en hoek) :
the size of the unknown sides and angles) :
1.1
ΔABC as / if ∠A = 49°, ∠B = 72°, BC = 23
1.2
ΔDEF as / if ∠D = 67°, ∠E = 38°, EF = 56
1.3
ΔKLM as / if ∠L = 29,5°, ∠M = 66,3°, KM = 13
1.4
ΔPQR as / if ∠P = 67,2°, ∠R = 38,8°, PQ = 45
1.5
ΔABC as / if ∠A = 59°, BC = 32, AC = 23
1.6
ΔDEF as / if ∠D = 74°, DE = 38, EF = 56
1.7
ΔKLM as / if ∠L = 138°, KL = 58, KM = 123
1.8
ΔPQR as / if ∠Q = 140°, QR = 48, PR = 74
Vraag / Question 2.
In die voorbeeld mag nie een van jou antwoorde 180° bevat nie. /
In this example not one of your answers may contain 180°
In die figuur, gee / In the diagram write down
2.1
q in terme van / in terms of r, x, y
2.2
r in terme van / in terms of q, x, y
2.3
p in terme van / in terms of q, x, y
2.4
p in terme van / in terms of r, x, y
2.5
q in terme van / in terms of p, x, y
2.6
r in terme van / in terms of p, x, y
Vraag / Question 3.
Gebruik die cos-reël om die onbekende te bepaal :
Use the cosine rule to determine the unknown :
3.1
In ΔABC bepaal / determine AC as / if AB = 8; BC = 7; ∠B = 53°
3.2
In ΔDEF bepaal / determine DF as / if DE = 11; EF = 5; ∠E = 144°
3.3
In ΔKLM bepaal / determine KM as / if KL = 4,2; LM = 3,7; ∠L = 130°
3.4
In ΔPQR bepaal / determine QR as / if QR = 632; PR = 543; ∠R = 56°
3.5
In ΔXYZ bepaal / determine YZ as / if XY = 98,3; XZ = 76,9; ∠X = 98,2°
3.6
In ΔABC bepaal / determine AB as / if AC = 9,6; BC = 4,7; ∠C = 144°
3.7
In ΔDEF bepaal / determine ∠D as / if DE = 10; EF = 5; DF = 14
3.8
In ΔKLM bepaal / determine ∠L as / if KL = 9; LM = 11; KM = 5
3.9
In ΔPQR bepaal / determine ∠P as / if PQ = 6; QR = 4; PR = 3
3.10
In ΔXYZ bepaal / determine ∠X as / if XY = 9; YZ = 11; XZ = 5
3.11
In ΔDEF bepaal / determine ∠E as / if DE = 10; EF = 5; DF = 14
3.12
In ΔPQR bepaal / determine ∠Q as / if p = 4,3; q = 6; r = 4
Vraag / Question 4.
Gebruik die oppervlakte-reël om die onbekende te bepaal :
Use the area rule to determine the unknown :
4.1
oppervlakte van / area of ΔABC as / if AC = 5 cm; BC = 8 cm; ∠C = 35°
4.2
oppervlakte van / area of ΔDEF as / if DE = 18 cm; DF = 24 cm; ∠D = 76°
4.3
oppervlakte van / area of ΔFGH as / if h = 14,3 m; f = 18,6 m; ∠G = 141°
4.4
oppervlakte van / area of ΔKLM as / if m = 3,1 m; l = 5,2 m; ∠K = 123°
4.5
oppervlakte van / area of ΔPQR as / if p = 8 cm; r = 5 cm; ∠Q = 72°
4.6
oppervlakte van / area of ΔXYZ as / if x = 21,3 mm; y = 15,8 cm; ∠Z = 48°
4.7
oppervlakte van / area of ΔABC as / if ∠B = 65°; ∠C = 43°; a = 13 cm
4.8
oppervlakte van / area of ΔDEF as / if ∠D = 38°; ∠F = 75°; d = 16 cm
4.9
oppervlakte van / area of ΔFGH as / if ∠H = 125°; ∠G = 32°; f = 9,5 dm
4.10
oppervlakte van / area of ΔKLM as / if ∠K = 23°; ∠L = 138°; m = 28,3 mm
4.11
In ΔPQR bepaal / determine q as / if ∠P = 61°; r = 54 cm en / and
die oppervlakte van / the area of ΔPQR = 685 cm2.
4.12
In ΔXYZ bepaal / determine x as / if ∠Z = 27°; y = 6 mm en / and
die oppervlakte van / the area of ΔXYZ = 9,5 mm2.
4.13
In ΔABC bepaal / determine c as / if ∠B = 123,2°; a = 9 cm en / and
die oppervlakte van / the area of ΔABC = 30,12 cm2.
4.14
In ΔDEF bepaal / determine d as / if ∠F = 28°; e = 28 cm en / and
die oppervlakte van / the area of ΔDEF = 168,53 cm2.
4.15
In ΔPQR bepaal / determine ∠P as / if q = 8 cm; r = 5 cm en / and
die oppervlakte van / the area of ΔPQR = 19 cm2.
4.16
In ΔFGH bepaal / determine ∠G as / if f = 14 cm; h = 19 cm en / and
die oppervlakte van / the area of ΔFGH = 83,7 cm2.
Vraag / Question 5.
In die voorbeeld mag nie een van jou antwoorde 180° bevat nie. /
In this example not one of your answers may contain 180°
In die voorbeeld word die oppervlakte van die driehoek gegee deur A. /
In this example the area of the triangle is represented by A.
In die figuur, gee / In the diagram write down
5.1
g in terme van / in terms of h, x, y
5.2
g in terme van / in terms of f, h, x
5.3
f in terme van / in terms of g, h, x, y
5.4
f in terme van / in terms of g, x, y
5.5
A in terme van / in terms of f, h, x
5.6
A in terme van / in terms of g, h, x, y
5.7
f in terme van / in terms of A, h, x
5.8
g in terme van / in terms of A, h, x, y
5.9
sin x in terme van / in terms of g, h, y
5.10
cos y in terme van / in terms of f, g, h
5.11
sin x in terme van / in terms of A, f, h
Vraag / Question 6.
In ΔKLM is KL = 5 eenhede, KM = 7 eenhede
en cos L = ½.
Bereken, sonder die gebruik van
'n sakrekenaar, die lengte van LM.
In ΔKLM KL = 5 units, KM = 7 units
and cos L = ½.
Calculate, without using a pocket calculator
the length of LM.
Vraag / Question 7.
In die figuur is NR ⊥ PR, PQ ⊥ QR,
∠PRQ = α, ∠MPR = β, ∠MRN = θ en QR = a.
In the figure NR ⊥ PR, PQ ⊥ QR,
∠PRQ = α, ∠MPR = β, ∠MRN = θ and QR = a.
Bewys dat / Prove that
a sin β
MR = ─────────────
cos α . cos (θ − β)
Vraag / Question 8.
In die figuur is ED = a eenhede, AB = AC,
∠ACB = θ, ∠ADC = α, ∠AEF = β,
∠FED = 90° en ED ⊥ BD
In the figure ED = a units, AB = AC,
∠ACB = θ, ∠ADC = α, ∠AEF = β,
∠FED = 90° and ED ⊥ BD
8.1
Bepaal AD in terme van α en β /
Determine AD in terms of α en β.
8.2
Bewys dat / Prove that AC = ─────────────
a . sin α . cos β
sin θ . sin (α − β)
Vraag / Question 9.
x
In die figuur is PQ = 2x eenhede, RQ = ────
sin θ
∠R = α, ∠P = (90° − θ)
x
In die figure PQ = 2x eenhede, RQ = ────
sin θ
∠R = α, ∠P = (90° − θ)
Bepaal α in terme van θ / Determine α in terms of θ
Vraag / Question 10.
In die figuur is AD = k eenhede, DB = h eenhede,
∠DCB = α, ∠ACB = β.
In the diagram AD = k units, DB = h units,
∠DCB = α, ∠ACB = β.
Bewys dat / Prove that
k . tan α = h(tan β − tan α)
Vraag / Question 11.
In ΔKLM is LM = x eenhede, ∠L = α, ∠M = β.
In ΔKLM LM = x units, ∠L = α, ∠M = β.
Toon aan dat
Show that
x . sin (α + β)
LM = ──────────
sin θ . sin β
Vraag / Question 12.
In die figuur is ∠ADB = 90°, ∠C = α,
∠ABD = β, AB = BC = x eenhede en DB ⊥ BC.
In the figure ∠ADB = 90°, ∠C = α,
∠ABD = β, AB = BC = x eenhede and DB ⊥ BC.
12.1
Bepaal / Determine DB in terme van / in terms of
x en / and α
12.2
Wys dat / Show that
AD2 = x2 (1 + tan2 α − 2 . tan α . cos β
Vraag / Question 13.
In die figuur is KL = LN = 10 cm,
NM = 6 cm, ∠KML = 38,1°, ∠N = 120°.
In the figure KL = LN = 10 cm,
NM = 6 cm, ∠KML = 38,1°, ∠N = 120°.
13.1
Bepaal / Determine LM
13.2
Bepaal die grootte van / Determine the size of ∠K.
13.3
Bereken die oppervlakte van KLNM. /
Calculate the area of KLNM.