Ngā Tauira o ngā Pātai me ngā Kōrero mō te Whakamāoritanga Pāngarau
Ko te whakamāoritanga he panonitanga āhuahanga e neke ana i ia pūwāhi i runga i tētahi papa i tētahi tawhiti motuhake i roto i tētahi ahunga motuhake. I roto i te pāngarau, he maha ngā wā ka whakamahia te whakamāoritanga hei neke i tētahi mea ki tētahi ahunga me te kore e whakarerekē i tōna āhua, i tōna aronga rānei. I roto i tēnei tuhinga, ka matapakihia e mātou ētahi tauira raruraru me ngā kōrero e pā ana ki te whakamāoritanga pāngarau hei āwhina i ngā kaipānui ki te mārama ake ki tēnei ariā.
Ngā Ariā Taketake o te Whakamāoritanga
Ka taea te whakaatu i te whakamāoritanga i roto i ngā taunga rua-ahu mā te whakamahi i te tuhi pūāhua. Mena ka whakamāoritia te pūwāhi A(x, y) e te pūāhua \((a, b)\), ka taea te tatau i te pūwāhi hua A' (\(x'\), \(y'\)) mā te whakamahi i te tātai:
\[ x' = x + a \]
\[ y' = y + b \]
Dimana:
– Ko te taunga tīmatanga ko \( (x, y) \)
– Ko te weke whakawhiti te \( (a, b) \)
– Ko \( (x', y') \) ngā taunga o te hua whakamāoritanga.
Ngā Pātai Tauira me te Kōrero
Anei ētahi tauira o ngā pātai e pā ana ki te whakamāoritanga pāngarau me ā rātou matapakinga:
Tauira Pātai 1:
Pātai:
Kei ngā taunga (3, 4) te Pūwāhi A. Whakamāoritia te pūwāhi A ki te whārite \( (5, -2) \). Whakatauhia ngā taunga hou o te pūwāhi A.
Kōrero:
E mōhiotia ana:
\[ \text{Ngā taunga taketake o te pūwāhi A} = (3, 4) \]
\[ \text{Te whakamāoritanga o te ira} = (5, -2) \]
Whakamahia te tātai whakamāori:
\[ x' = x + a \]
\[ y' = y + b \]
Whakakapia ngā uara:
\[ x' = 3 + 5 = 8 \]
\[ y' = 4 + (-2) = 2 \]
Nō reira, ko ngā taunga hou o te pūwāhi A i muri i te whakawhiti ko \( (8, 2) \).
Tauira Pātai 2:
Pātai:
He tapatoru ngā taunga pūwāhi \( A(1, 2) \), \( B(3, 5) \), me \( C(6, 1) \). Whakamāoritia te tapatoru me te whārite \( (-2, 4) \). Whakatauhia ngā taunga pūwāhi o te tapatoru i muri i te whakamāoritanga.
Kōrero:
E mōhiotia ana ngā taunga o te tihi o te tapatoru me te wekere whakawhiti.
Ngā taunga o te pūwāhi A':
\[ x' = 1 + (-2) = -1 \]
\[ y' = 2 + 4 = 6 \]
Kātahi, \( A' = (-1, 6) \).
Ngā taunga o te pūwāhi B':
\[ x' = 3 + (-2) = 1 \]
\[ y' = 5 + 4 = 9 \]
Kātahi, \( B' = (1, 9) \).
Ngā taunga o te pūwāhi C':
\[ x' = 6 + (-2) = 4 \]
\[ y' = 1 + 4 = 5 \]
Kātahi, \( C' = (4, 5) \).
Nō reira, i muri i te whakamāoritanga, ko ngā taunga o ngā pito o te tapatoru hou ko \( A'(-1, 6) \), \( B'(1, 9) \), me \( C'(4, 5) \).
Tauira Pātai 3:
Pātai:
I hoatu te pūwāhi P i ngā taunga \( (-3, 0) \). Whakatauhia te hua o te whakawhiti o te pūwāhi P e te whārite \( (7, -5) \).
Kōrero:
I runga i ngā taunga o P me te wekere whakawhiti.
Whakamahia te tātai whakamāori:
\[ x' = x + a \]
\[ y' = y + b \]
Whakakapia ngā uara:
\[ x' = -3 + 7 = 4 \]
\[ y' = 0 + (-5) = -5 \]
Nō reira, ko ngā taunga o te hua whakawhiti o te pūwāhi P ko \( (4, -5) \).
Tauira Pātai 4:
Pātai:
Kei ngā taunga te Pūwāhi Q ((4, -3)). Mēnā ka whakawhitia te pūwāhi Q kia noho ngā taunga hou hei (9, 1)), whakatauhia te whārite whakawhiti i whakamahia.
Kōrero:
E mōhiotia ana:
\[ \text{Ngā taunga tīmatanga} = (4, -3) \]
\[ \text{Ngā taunga hua} = (9, 1) \]
Whakamahia te tātai whakamāori hei kimi i te whārite \( (a, b) \):
\[ x' = x + a \]
\[ y' = y + b \]
E mōhiotia ana ngā hua o te whakamāoritanga:
\[ 9 = 4 + a \]
\[ 1 = -3 + b \]
Nō reira, ka taea te tatau i te whakamāoritanga o te wetere penei:
\[ a = 9 – 4 = 5 \]
\[ b = 1 + 3 = 4 \]
Nō reira, ko te wekere whakawhiti i whakamahia ko \( (5, 4) \).
Tauira Pātai 5:
Pātai:
He pūwāhi kokonga kei te tapawhā ABCD \( A(1, 2) \), \( B(1, 5) \), \( C(4, 5) \), me \( D(4, 2) \). Whakamāoritia te tapawhā ki te whārite \( (3, -1) \). Whakatauhia ngā taunga hou o te tapawhā ABCD.
Kōrero:
E mōhiotia ana:
\[ \text{Te whakamāoritanga o te ira} = (3, -1) \]
Ngā taunga o te pūwāhi A':
\[ x' = 1 + 3 = 4 \]
\[ y' = 2 + (-1) = 1 \]
Kātahi, \( A' = (4, 1) \).
Ngā taunga o te pūwāhi B':
\[ x' = 1 + 3 = 4 \]
\[ y' = 5 + (-1) = 4 \]
Kātahi, \( B' = (4, 4) \).
Ngā taunga o te pūwāhi C':
\[ x' = 4 + 3 = 7 \]
\[ y' = 5 + (-1) = 4 \]
Kātahi, \( C' = (7, 4) \).
Ngā taunga o te pūwāhi D':
\[ x' = 4 + 3 = 7 \]
\[ y' = 2 + (-1) = 1 \]
Kātahi, \( D' = (7, 1) \).
Nō reira, ko ngā taunga hou o te tapawhā ABCD i muri i te whakawhiti ko \( A'(4, 1) \), \( B'(4, 4) \), \( C'(7, 4) \), me \( D'(7, 1) \).
Whakamutunga
He tino taketake te whakawhiti engari he mea nui te whakawhiti i roto i te āhuahanga. Mā te mōhio ki tēnei tikanga ka taea e tātou te mahi i ngā mahi āhuahanga, pērā i te neke i ngā mea me te kore e whakarerekē i ō rātou āhua, i ō rātou rahi rānei.
Mā te mārama ki te ariā o te whakamāoritanga mā roto i ngā tauira raruraru kua kōrerotia i runga ake nei, ko te tumanako ka taea e ngā kaipānui te mārama ake me te whakamahi i tēnei ariā i roto i ngā raruraru maha me te ao tūturu. He whai hua te whakamāoritanga ehara i te mea i roto i te pāngarau anake engari i roto hoki i ngā momo mara rerekē, tae atu ki te ahupūngao, te whakairoiro rorohiko, me te hoahoa.
