Ngā Tauira Pātai e Matapaki ana i te Tūnga o tētahi Pūwāhi e Pā Ana ki tētahi Porowhita
He kaupapa nui te whakatau i te tūnga o tētahi pūwāhi e pā ana ki tētahi porowhita i roto i te āhuahanga taketake, inā koa i roto i te ako i ngā porowhita. I roto i tēnei tuhinga, ka matapakihia e mātou ētahi tauira raruraru e pā ana ki te tūnga o tētahi pūwāhi e pā ana ki tētahi porowhita, me ō rātou whakamārama. Mā tēnei ka āwhina i te whakamārama i te ariā mā roto i ngā tono mahi.
Pendahuluan
I mua i te ruku ki ngā tauira pātai, me mahara tātou ki ngā tūranga e toru o tētahi pūwāhi i runga i tētahi porowhita:
1. I roto i te porowhita: Mena he iti ake te tawhiti o tētahi pūwāhi ki te pokapū o te porowhita i te pūtoro o te porowhita.
2. I waho o te porowhita: Mena he nui ake te tawhiti o te pūwāhi ki te pokapū o te porowhita i te pūtoro o te porowhita.
3. I runga i te porowhita: Mena he rite te tawhiti mai i te pūwāhi ki te pokapū o te porowhita ki te pūtoro o te porowhita.
Mā te pāngarau, ka taea te whakatau i te tūranga o te pūwāhi \(T(x_1, y_1)\) e pā ana ki te porowhita e pokapū ana i \((a, b)\) me te radius \(r\) mā te whakataurite i \(T(x_1, y_1)\) ki te whārite o te porowhita, arā:
\[
(x – a)^2 + (y – b)^2 = r^2
\]
Mena ka puta te uara mai i te hua o te whakakapinga o \(x_1\) me \(y_1\) ki te whārite:
– He iti iho i te \(r^2\), kei roto te pūwāhi i te porowhita.
– Ki te nui ake i te \(r^2\), kei waho o te porowhita te pūwāhi.
– Pērā i te \(r^2\), kei runga i te porowhita te pūwāhi.
Ngā Pātai Tauira me te Kōrero
Pātai 1
Whakatauhia te tūranga o te pūwāhi \(T(3, 4)\) e pā ana ki te porowhita me te whārite \( (x – 1)^2 + (y – 2)^2 = 25 \).
Kōrero:
Ko te taahiraa tuatahi ko te aromatawai i te whārite o te porowhita, me te kimi i te tawhiti mai i te pūwāhi \(T(3, 4)\) ki te pokapū o te porowhita \((1, 2)\).
1. Tāutuhia te pokapū me te pūtoro o te porowhita:
Whārite o te porowhita: \( (x – 1)^2 + (y – 2)^2 = 25 \)
– Te pokapū o te porowhita (\(a, b\)): (1, 2)
– Te pūtoro o te porowhita (\(r\)): \(\sqrt{25} = 5\)
2. Tātaihia te tawhiti i waenganui i te pūwāhi \(T(3, 4)\) me te pokapū o te porowhita \( (1, 2) \):
\[
D = \sqrt{(3 – 1)^2 + (4 – 2)^2} = \sqrt{2^2 + 2^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}
\]
Ko te uara \( 2\sqrt{2} \approx 2 \times 1.414 = 2.828 \) (iti iho i te \(5\)).
3. Whakamutunga:
Nā te mea ko \( 2\sqrt{2} < 5 \), ko te pūwāhi \( T(3, 4) \) kei roto i te porowhita. Pātai 2 He pokapū kei te pūwāhi \( (0, 0) \) o te porowhita, ā, ko te radius he 7. Whakatauhia te tūranga o te pūwāhi \(P(5, 6)\) e pā ana ki te porowhita.
Pātai 3
Whakatauhia te tūranga o te pūwāhi \(M(2, -1)\) e pā ana ki te porowhita me te whārite \( x^2 + y^2 = 5 \).
Kōrero:
1. Tātaihia te tawhiti mai i te pūwāhi \(M(2, -1)\) ki te pokapū o te porowhita \( (0, 0) :
\[
D = \sqrt{(2 – 0)^2 + (-1 – 0)^2} = \sqrt{4 + 1} = \sqrt{5}
\]
2. Whakatauritea te tawhiti \(D\) ki te pūtoro o te porowhita:
Te pūtoro o te porowhita (\(r\)) = \(\sqrt{5}\).
3. Whakamutunga:
Nā te mea ko \( \sqrt{5} = \sqrt{5} \), ko te pūwāhi \( M(2, -1) \) kei runga i te porowhita.
Pātai 4
He porowhita kei waenganui i te \( (4, 3) \) he radius \(\sqrt{10}\). Whakaaturia te tūranga o te pūwāhi \( N(7, 7) \) e pā ana ki tēnei porowhita.
Kōrero:
1. Whārite Porowhita:
Ko te whārite o tētahi porowhita me te pokapū \( (4, 3) \) me te radius \( \sqrt{10} \) ko:
\[
(x – 4)^2 + (y – 3)^2 = 10
\]
2. Tātaihia te tawhiti mai i te pūwāhi \( N(7, 7) \) ki te pokapū o te porowhita \( (4, 3) \):
\[
D = \sqrt{(7 – 4)^2 + (7 – 3)^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5
\]
3. Whakamutunga:
Nā te mea ko \( 5 > \sqrt{10} \), kei waho o te porowhita te pūwāhi \( N(7, 7) \).
Te Katinga
Mā te mārama ki te tatau i te tawhiti o tētahi pūwāhi mai i te pokapū o tētahi porowhita, me te whakatairite i taua pūwāhi ki te radius, ka taea e tātou te whakatau ngāwari i te tūranga o tētahi pūwāhi e pā ana ki te porowhita. Ko te matapakinga o tēnei tuhinga e tumanakohia ana kia mārama te ariā me te whakaoti rapanga e pā ana ki te tūranga o tētahi pūwāhi e pā ana ki tētahi porowhita.
I roto i te mahi, he tino whai hua te mōhio ki te tūranga o ēnei pūwāhi i roto i ngā momo mahi pāngarau, tae atu ki te tātari āhuahanga, te hoahoa whakairoiro, me te hangarau. Nō reira, ko te mōhio ki tēnei ariā he tūāpapa nui e tika ana kia aro nuihia, kia māramahia hoki.
