WISKUNDE
GRAAD 10
NOG OEFENINGE
Lengte en middelpunte van lyne : antwoorde.
GRAAD 10
NOG OEFENINGE
Lengte en middelpunte van lyne : antwoorde.
MATHEMATICS
GRADE 10
MORE EXERCISES
Lengths and midpoints of lines : answers.
GRADE 10
MORE EXERCISES
Lengths and midpoints of lines : answers.
Die lengte van lyn AB word gegee deur d(AB) en
die middelpunt deur M(AB).
The length of line AB is represented by d(AB) and
the midpoint by M(AB).
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1.1 d(AB) = √((x2 − x1)2 + (y2 − y1)2)
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= √((6 − 2)2 + (11 − 3)2)
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= √(4 )2 + (8)2)
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= √80 = 8,94
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d(CD) = √((x2 − x1)2 + (y2 − y1)2)
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= √((6 − 3)2 + (2 − 11)2)
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= √9 + 81
__
= √90 = 9,49
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d(AB) = √((x2 − x1)2 + (y2 − y1)2)
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= √((−3 − (−8))2 + (3 − (−5))2)
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= √(5 )2 + (8)2)
__
= √89 = 9,43
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d(PQ) = √((xQ − xP)2 + (yQ − yP)2)
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= √((−6 − (−10))2 + (−8 − (−2))2)
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= √16 + 36
__
= √52 = 7,21
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d(RS) = √((xS − xR)2 + (yS − yR)2)
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= √((−1 − (−7))2 + (6 − (−2))2)
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= √(6 )2 + (8)2)
__
= √100 = 10
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d(AB) = √((xB − xA)2 + (yB − yA)2)
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= √((3 − (−6))2 + (6 − (−2))2)
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= √81 + 64
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= √145 = 12,04
2.1 d(DE) = √80 ∴ DE2 = (√80)2
((xE − xD)2 + (yE − yD)2) = (√80)2
((e − (−4))2 + (−19 − (−11))2) = (√80)2
(e + 4)2 + (−8)2) = 80
e2 + 8e +16 + 64 −80 = 0
e2 + 8e = 0
e(e + 8) = 0
e = 0 of / or e = −8
2.3 d(KL) = 11,662 ∴ KL2 = 11,6622
((xL − xK)2 + (yL − yK)2) = 11,6622
((10 − k)2 + (−4 − 62) = 11,6622
(10 − k)2 + (−10)2) = 136
100 − 20k + k2 + 100 −136 = 0
k2 − 20k + 64 = 0
(k − 4)(k − 16) = 0
k = 4 of / or k = 16
2.5 d(RS) = √18 ∴ RS2 = (√18)2
((xs − xR)2 + (yS − yR)2) = (√18)2
((−3 − (−6))2 + (5 − r)2) = (√18)2
32 + (5 − r)2 = 18
9 + 25 − 10r + r2 − 18 = 0
r2 − 10r + 16 = 0
(r − 2)(r − 8) = 0
r = 2 of / or r = 8
((xE − xD)2 + (yE − yD)2) = (√80)2
((e − (−4))2 + (−19 − (−11))2) = (√80)2
(e + 4)2 + (−8)2) = 80
e2 + 8e +16 + 64 −80 = 0
e2 + 8e = 0
e(e + 8) = 0
e = 0 of / or e = −8
2.3 d(KL) = 11,662 ∴ KL2 = 11,6622
((xL − xK)2 + (yL − yK)2) = 11,6622
((10 − k)2 + (−4 − 62) = 11,6622
(10 − k)2 + (−10)2) = 136
100 − 20k + k2 + 100 −136 = 0
k2 − 20k + 64 = 0
(k − 4)(k − 16) = 0
k = 4 of / or k = 16
2.5 d(RS) = √18 ∴ RS2 = (√18)2
((xs − xR)2 + (yS − yR)2) = (√18)2
((−3 − (−6))2 + (5 − r)2) = (√18)2
32 + (5 − r)2 = 18
9 + 25 − 10r + r2 − 18 = 0
r2 − 10r + 16 = 0
(r − 2)(r − 8) = 0
r = 2 of / or r = 8
2.2 d(FG) = √41
∴ FG2 = (√41)2
((xG − xF)2 + (yG − yF)2) = (√41)2
((8 − 3)2 + (g − 8)2) = (√41)2
52 + (g − 8)2 = 41
25 + g2 − 16g + 64 −41 = 0
g2 − 16g + 48 = 0
(g − 4)(g − 12) = 0
g = 4 of / or g = 12
2.4 d(PQ) = 7,81 ∴ PQ2 = 7,812
((xQ − xP)2 + (yQ − yP)2) = 7,812
((−2 − (−7))2 + (−2 − p)2) = 7,812
52 + (−(p + 2))2 = 61
25 + p2 + 4p + 4 − 61 = 0
p2 + 4p − 32 = 0
(p − 4)(p + 8) = 0
p = 4 of / or p = −8
2.6 d(AB) = √13 ∴ AB2 = (√13)2
((xB − xA)2 + (yB − yA)2) = (√13)2
((b − 8)2 + (8 − 10)2) = (√13)2
(b − 8)2 + (−2)2 = 13
b2 − 16b + 64 + 4 −13 = 0
b2 − 16b + 55 = 0
(b − 5)(b − 11) = 0
b = 5 of / or b = 11
((xG − xF)2 + (yG − yF)2) = (√41)2
((8 − 3)2 + (g − 8)2) = (√41)2
52 + (g − 8)2 = 41
25 + g2 − 16g + 64 −41 = 0
g2 − 16g + 48 = 0
(g − 4)(g − 12) = 0
g = 4 of / or g = 12
2.4 d(PQ) = 7,81 ∴ PQ2 = 7,812
((xQ − xP)2 + (yQ − yP)2) = 7,812
((−2 − (−7))2 + (−2 − p)2) = 7,812
52 + (−(p + 2))2 = 61
25 + p2 + 4p + 4 − 61 = 0
p2 + 4p − 32 = 0
(p − 4)(p + 8) = 0
p = 4 of / or p = −8
2.6 d(AB) = √13 ∴ AB2 = (√13)2
((xB − xA)2 + (yB − yA)2) = (√13)2
((b − 8)2 + (8 − 10)2) = (√13)2
(b − 8)2 + (−2)2 = 13
b2 − 16b + 64 + 4 −13 = 0
b2 − 16b + 55 = 0
(b − 5)(b − 11) = 0
b = 5 of / or b = 11
Gebruik die formule om die middelpunt te bereken.
Use the formula to determine the midpoint.
−4 + p −7 + q
────── = −1 ; ────── = −2
2 2
−4 + p = 2 × −1 −7 + q = 2 × −2
p = −2 + 4 q = −4 + 7
p = 2 q = 3
p + (−1) q + (−21)
─────── = −5 ; ─────── = −17
2 2
p − 1 = 2 × −5 q − 21 = 2 × −17
p = −10 + 1 q = −34 + 21
p = −9 q = −13
p + 8 q + 14
────── = 5 ; ────── = 10
2 2
p + 8 = 2 × 5 q + 14 = 2 × 10
p = 10 − 8 q = 20 − 14
p = 2 q = 6
3 + p 5 + q
─────── = 5,5 ; ─────── = 7,5
2 2
p + 3 = 2 × 5,5 q + 5 = 2 × 7,5
p = 11 − 3 q = 15 − 5
p = 8 q = 10
−10 + p −8 + q
────── = −7,5 ; ────── = −4,5
2 2
p − 10 = 2 × (−7,5) q − 8 = 2 × −4,5
p = −15 + 10 q = −9 + 8
p = −5 q = −1
p + 3 q + 5
─────── = −1 ; ─────── = 1
2 2
p + 3 = 2 × (−1) q + 5 = 2 × 1
p = −2 − 3 q = 2 − 5
p = −5 q = −3
p + 7 8 + q
────── = 2 ; ────── = 10
2 2
p + 7 = 2 × 2 q + 8 = 2 × 10
p = 4 − 7 q = 20 − 8
p = −3 q = 12
−7 + p q + (−4)
─────── = −1 ; ─────── = 2
2 2
p − 7 = 2 × (−1) q − 4 = 2 × 2
p = −2 + 7 q = 4 + 4
p = 5 q = 8
As die lyn AB sy eie lengte na C verleng word, is
B die middelpunt van die lyn AC. Gebruik die
formule om die endpunt te bereken.
B die middelpunt van die lyn AC. Gebruik die
formule om die endpunt te bereken.
If the line AB is produced its own length to C,
then B is the midpoint of the line AC. Use
the formula to determine the other point.
then B is the midpoint of the line AC. Use
the formula to determine the other point.
2 + p 3 + q
────── = 6 ; ────── = 8
2 2
p + 2 = 2 × 6 q + 3 = 2 × 8
p = 12 − 2 q = 16 − 3
p = 10 q = 13
C is die punt / is the point (10 ; 13)
−8 + p 12 + q
─────── = −5 ; ─────── = 7
2 2
p − 8 = 2 × −5 q + 12 = 2 × 7
p = −10 + 8 q = 14 − 12
p = −2 q = 2
F is die punt / is the point (−2 ; 2)
6 + p 9 + q
────── = −1 ; ────── = 1
2 2
p + 6 = 2 × −1 q + 9 = 2 × 1
p = −2 − 6 q = 2 − 9
p = −8 q = −7
M is die punt / is the point (−8 ; −7)
−10 + a −2 + b
─────── = −3 ; ─────── = −8
2 2
p − 8 = 2 × −5 q + 12 = 2 × 7
p = −10 + 8 q = 14 − 12
p = −2 q = 2
R is die punt / is the point (4 ; −14)