WISKUNDE
GRAAD 12
NOG OEFENINGE
Oplos van 2D en 3D figure.
MATHEMATICS
GRADE 12
MORE EXERCISES
Solving 2D and 3D figures.
Vraag / Question 1
In meegaande diagram is die oppervlakte van
ΔTKM gelyk aan die oppervlakte van ΔKML.
∠KML = θ; θ is 'n stomphoek en ML = 18.
1.1
Bepaal die oppervlakte van ΔTKM.
14
1.2
Toon aan dat sin θ = ───
15
1.3
Bepaal die waarde van θ, korrek tot die naaste graad.
In the diagram above the area of ΔTKM is equal to the area of ΔKML. ∠KML = θ; θ is
an obtuse angle and ML = 18.
1.1
Determine the area of ΔTKM.
14
1.2
Show that sin θ = ───
15
1.3
Determine the value of θ, correct to the nearest grade.
Vraag / Question 2
In meegaande diagram is AB ⊥ BC,
CD = a. Bewys dat
In the diagram AB ⊥ BC,
CD = a. Prove that
a ⋅ sin (x + y) ⋅ cos z
BC = ─────────────
sin x
Vraag / Question 3
In meegaande diagram is AE || DC,
∠ADC = 90°, ∠EAC = y, ∠EAB = x
en ∠ABC = w + x. Bewys dat
In the diagram AE || DC,
∠ADC = 90°, ∠EAC = y, ∠EAB = x
and ∠ABC = w + x. . Prove that
AB ⋅ sin (x + x)
AC = ───────────
sin (y + w)
Vraag / Question 4
In die diagram / In the diagram
AB = 400 m, ∠DAC = 59°, ∠CAB = 46°,
∠ABD = 29°, ∠DBC = 55°.
Vind CD en die hoek wat AB met CD maak.
Find CD and the angle that AB forms with CD.
Vraag / Question 5
In die diagram / In the diagram
AF = p, ∠CAB = x, ∠FAB = y,
∠CFE = w, EF || AB.
Bewys dat / Prove that
p ⋅ sin (x − y) ⋅ sin (w − y)
AD = ──────────────────
sin (w − x) ⋅ cos y
Vraag / Question 6
In die diagram / In the diagram
CB = h, ∠CAB = x, ∠DAB = y,
∠CDE = w, ∠CED = 90°, ∠B = 90°.
Bewys dat / Prove that
h . (sin (x − y) + sin (w − x))
AD + DC = ─────────────────────
sin x . sin (w − x)
Vraag / Question 7
In die diagram / In the diagram
PQ = a, QR = b, ∠QPR = y,
∠SPR = x, ∠SRP = w.
Bewys dat / Prove that
a . sin x
SR = ─────────────
sin (w + x) . cos y
Vraag / Question 8
In die diagram / In the diagram
AD = h, BC = d, ∠ABD = x,
∠DCB = y, ∠DBC = Z, AD ⊥ BD.
B, C en D is in dieselfde horisontale vlak.
B, C and D are in the same horizontal plane.
Bewys dat / Prove that
d . sin y . tan x
h = ─────────────
sin (y + z)
Vraag / Question 9
In die diagram / In the diagram
Q, R en S is punte in dieselfde horisontale vlak
en PQ is loodreg op die vlak.
Q, R and S are points in the same horizontal
plane and PQ is perpendicular to the plane.
QS = a, ∠QRS = x, ∠QSR = y, ∠PRQ = z.
Bewys dat / Prove that
a . sin y . tan z
PQ = ─────────────
sin x
Vraag / Question 10
In die diagram / In the diagram
is A, B en C punte in dieselfde horisontale vlak
en TC is loodreg op die vlak.
A, B and C are points in the same horizontal
plane and TC is perpendicular to the plane.
AB = d, ∠TAC = x, ∠TAB = y, ∠TBA = z.
Bewys dat / Prove that
d . sin z . sin x
TC = ─────────────
sin (y + z)
Vraag / Question 11
In die diagram / In the diagram
is A, B en C punte in dieselfde horisontale vlak
en DA is loodreg op die vlak.
A, B and C are points in the same horizontal
plane and DA is perpendicular to the plane.
DA = h, AC = BC = d , ∠ABC = x,
∠BAC = y, ∠DBA = z.
———————
Bewys dat / Prove that h = d . (√(2 + 2 cos (x + y))) . tan z
Vraag / Question 12
In die diagram / In the diagram
is B, D en C punte in dieselfde horisontale vlak
en AD is loodreg op die vlak.
B, D and C are points in the same horizontal
plane and AD is perpendicular to the plane.
AD = h, ∠ABD = x, ∠ABC = y
en / and ∠BDC = 90°
tan x
Bewys dat / Prove that tan DBC = ─────
tan y
Vraag / Question 13
ABD is 'n driehoekige kamp met ∠D = 90°.
C is 'n punt binne die kamp en punt D is reg
noord van B. Verder is ∠ DBA = x,
∠ MCA = y, ∠DBC = z en AB = a.
ABD is a triangular field with ∠D = 90°.
C is a point inside the field and point D is
directly north of B. Furthermore ∠ DBA = x,
∠ MCA = y, ∠DBC = z and AB = a.
13.1
Bepaal ∠BAC in terme van x en y. / Determine ∠BAC in terms of x and y.
13.2
Druk ∠ACB uit in terme van y en z. / Express ∠ACB in terms of y and z.
a sin (y − x)
13.3
Bewys dat / Prove that CB = ─────────
sin (y − z)
[ O.V.S.Vraestel 2 1991 / O.F.S. Question paper 2 1991 ]
Vraag / Question 14
In die figuur is MLN 'n horisontale vlak met
∠LMN = ∠LNM = x. KL is 'n vertikale toring
by L en die dieptehoek vanaf K, die top van die
toring, na M is 2x. Die hoogte van die
toring is h meter.
In the accompanying figuur MLN is a horizontal
plane with ∠LMN = ∠LNM = x. KL is a vertical
tower at L and the angle of depression from K,
the top of the tower, to M is 2x. The height of the tower is h metres.
h cos 2x
Bewys dat / Prove that MN = ───────
sin x
[ G.M.R.Vraestel 2 (HG) 1991 / J.M.B. Question paper 2 (HG) 1991 ]
Vraag / Question 15
In die figuur stel PQ 'n vertikale
netbalpaal voor. Twee spelers het stelling
ingeneem by punte A en B op die grens
van die netbalbaan sodat A, Q en B
in 'n reguitlyn lê. 'n Derde speler bevind
haar by punt C binne die baan. Die
hoogtehoeke van P vanaf A en C is x, terwyl
die hoogtehoek van P vanaf B y is. Die lengte van QB is k.
In the figure, PQ represents a vertical netball pole. Two players are positioned at points A and B
on the boundary of the netball court so that A, Q and B lie in a straight line. A third player is
positioned at a point C inside the court. The angles of elevation of P from A and C are x, while the
angle of elevation of P from B is y. The length of QB is k.
k tan y
15.1
Bewys dat / Show that CQ = ──────
tan x
15.2
Bereken CQ as / Calculate CQ if k = 4,8 m, x = 42° en / and y = 32°.
15.3
Bereken nou die afstand tussen A en C as ∠CQB = 100°. /
Now calculate the distance between A and C if ∠CQB = 100°.
[ Raad van Afgevaardigdes Vraestel 2 (HG) 1991 / House of Delegates Question paper 2 (HG) 1991 ]