Nā nīnau hoʻohālike e kūkākūkā ana i ka wehewehe ʻana o kahi pōʻai
ʻO ka pōʻai kekahi o nā kinona geometric kumu a mākou e ʻike pinepine ai i ke ola o kēlā me kēia lā. I ka makemakika, he mau wehewehena a me nā ʻano kūikawā ko nā pōʻai. E ʻimi hohonu kēia ʻatikala i ka wehewehe ʻana o kahi pōʻai a me kona mau mea pili, a e hāʻawi pū i kekahi mau pilikia hoʻohālike a me kā lākou mau hoʻonā e hoʻonui i ko mākou ʻike i nā pōʻai.
Wehewehena o ka Pōʻai
ʻO ka pōʻai ka hui o nā kiko āpau i loko o kahi mokulele e like ka mamao mai kahi kiko paʻa i kapa ʻia ʻo ke kikowaena o ka pōʻai. ʻO ka mamao ma waena o ke kikowaena a me kekahi kiko ma ka pōʻai ua kapa ʻia ʻo ka radius o ka pōʻai. ʻO ka hoʻohālikelike laulā o kahi pōʻai me ke kikowaena ma kahi \((h, k)\) a me ka radius \(r\) i hāʻawi ʻia e:
\[ (x – h)^2 + (y – k)^2 = r^2 \]
Aia:
– ʻO \((h, k)\) nā hoʻonohonoho o ke kikowaena o ka pōʻai,
– ʻO \(r\) ka radius o ka pōʻai,
– ʻO \(x\) a me \(y\) nā hoʻonohonoho o kekahi kiko ma ka pōʻai.
Nā ʻElemene o kahi Pōʻai
Ma mua o ko kākou komo ʻana i nā nīnau hoʻohālike, he mea maikaʻi ke ʻike i kekahi o nā mea nui o ka pōʻai:
1. Kikowaena o ka Pōʻai: He kiko paʻa ke kikowaena o nā kiko a pau i like ka mamao.
2. Radius (r): Ka mamao mai ke kikowaena o ka pōʻai a i kekahi kiko ma ka pōʻai.
3. Anawaena (d): He laina pololei e hele ana ma ke kikowaena o ka pōʻai a hoʻohui i ʻelua mau kiko ma ka pōʻai, he pālua ka lōʻihi o ka radius (\(d = 2r\)).
4. Piʻo: ʻO ka ʻāpana o ke anapuni o kahi pōʻai e waiho ana ma waena o ʻelua mau kiko ma ka pōʻai.
5. Chord: He laina pololei e hoʻohui i ʻelua mau kiko ma kahi pōʻai akā ʻaʻole e hele ma waena o ke kikowaena.
6. Apothem: ʻO ka mamao pōkole loa mai ke kikowaena o kahi pōʻai a i kahi chord.
7. Ke kihi waena: Ke kihi i hoʻokumu ʻia e ʻelua mau radius mai ke kikowaena o ka pōʻai.
8. Ke kihi Perimeter: Ke kihi i hoʻokumu ʻia e ʻelua mau kaula e hui ana ma kekahi kiko ma kahi pōʻai.
Nā nīnau hoʻohālike a me nā kūkākūkā
Laʻana Nīnau 1
Nīnau: Hāʻawi ʻia kahi pōʻai me ke kikowaena ma kahi kiko \((3, 4)\) a ke hele nei ma waena o kahi kiko \((7, 4)\). E hoʻoholo i ka hoohalike o ka pōʻai.
Kūkākūkā:
No ka loaʻa ʻana o ke kaulike o kahi pōʻai, pono mākou e ʻike mua i kona radius. ʻOiai ke hele nei ka pōʻai ma ke kiko \((7, 4)\), hiki iā mākou ke helu i ka mamao ma waena o kēia kiko a me ke kikowaena o ka pōʻai, ʻo ia hoʻi \((3, 4)\).
\[
r = \sqrt{(x_2 – x_1)^2 + (y_2 – y_1)^2}
\]
\[
r = \sqrt{(7 – 3)^2 + (4 – 4)^2}
\]
\[
r = \sqrt{4^2 + 0^2}
\]
\[
r = 4
\]
Me ke kikowaena ma \((3, 4)\) a me ka radius 4, ʻo ke kaulike o ka pōʻai penei:
\[
(x – 3)^2 + (y – 4)^2 = 4^2
\]
\[
(x – 3)^2 + (y – 4)^2 = 16
\]
Laʻana Nīnau 2
Nīnau: E hoʻoholo i ka ʻāpana a me ke anapuni o kahi pōʻai me ka radius o 5 kenimika.
Kūkākūkā:
– Hiki ke helu ʻia ka ʻāpana o kahi pōʻai (A) me ka hoʻohana ʻana i ke ʻano hana \(A = \pi r^2\),
\[
A = \pi \times 5^2
\]
\[
A = 25\pi \text{ cm}^2
\]
Inā \(\pi \approx 3.14\), a laila:
\[
A \approx 25 \times 3.14 = 78.5 \text{ cm}^2
\]
– Hiki ke helu ʻia ke anapuni o kahi pōʻai (C) me ka hoʻohana ʻana i ke ʻano hana \(C = 2\pi r\),
\[
C = 2 manawa \pi manawa \5
\]
\[
C = 10\pi \text{ kenimika}
\]
Inā \(\pi \approx 3.14\), a laila:
\[
C \approx 10 \times 3.14 = 31.4 \text{ kenimika}
\]
Laʻana Nīnau 3
Nīnau: He kikowaena ko ka pōʻai ma ke kiko O a he 7 kenimika ka radius. Inā ua kaha ʻia kahi kaula o ka lōʻihi he 10 kenimika ma ka pōʻai, e hoʻoholo i ka mamao pōkole loa mai ke kikowaena O a i ke kaula.
Kūkākūkā:
No ka loaʻa ʻana o ka mamao pōkole loa mai ke kikowaena a i ke kaula, hoʻohana mākou i ke kumumanaʻo o ka apothem, ʻo ia ka mamao pōkole loa mai ke kikowaena a i ke kaula. Me ka radius o 7 cm a me ka lōʻihi o ke kaula he 10 cm, hiki iā mākou ke hoʻokokoke i kēia pilikia me ka hoʻohana ʻana i ka huinakolu ʻākau i hana ʻia.
Inā ʻo ke kikowaena o ke kaula he kiko C, a laila ʻo OC ka apothem a mākou e ʻimi nei. Inā ʻo A a me B nā hopena o ke kaula, a laila ʻo AC a me BC he 5 cm kēlā me kēia (hapalua o 10 cm).
Mai ka huinakolu OAC i ʻākau ʻia ma C, hoʻohana mākou i ka manaʻo Pythagorean:
\[
OA^2 = OC^2 + AC^2
\]
\[
7^2 = OC^2 + 5^2
\]
\[
49 = OC^2 + 25
\]
\[
OC^2 = 49 – 25
\]
\[
OC^2 = 24
\]
\[
OC = \sqrt{24} = 2\sqrt{6} \text{ kenimika}
\]
Laʻana Nīnau 4
Nīnau: Hāʻawi ʻia kahi pōʻai me ka hoohalike \(x^2 + y^2 + 6x – 8y + 9 = 0\). E hoʻoholo i ke kikowaena a me ka radius o ka pōʻai.
Kūkākūkā:
No ka loaʻa ʻana o ke kikowaena a me ka radius, hoʻololi mākou i ka hoohalike i ke ʻano maʻamau:
\[
x^2 + y^2 + 6x – 8y + 9 = 0
\]
Hoʻoponopono mākou ma ka hoʻomaikaʻi ʻana i ke ʻano quadratic:
\[
x^2 + 6x + y^2 – 8y = -9
\]
Ke hoʻohui a me ka unuhi ʻana i ka (6/2)\(^2\) i ka huaʻōlelo x a me ka (8/2)\(^2\) i ka huaʻōlelo y:
\[
x^2 + 6x + 9 + y^2 – 8y + 16 = -9 + 9 + 16
\]
\[
(x + 3)^2 + (y – 4)^2 = 16
\]
Mai ke kaulike \((x + 3)^2 + (y – 4)^2 = 16\), ʻike mākou ʻo ke kikowaena o ka pōʻai ʻo \((-3, 4)\) a ʻo ka radius ʻo \(r = \sqrt{16} = 4\).
Ka hopena
He manaʻo nui a koʻikoʻi ka pōʻai i ke geometry. Ma o nā pilikia hoʻohālike a me nā kūkākūkā ma luna, hiki iā mākou ke loaʻa kahi ʻike hohonu o nā ʻano like ʻole a me nā waiwai o nā pōʻai. ʻO ka hoʻomaopopo ʻana i kēia kumuhana e maʻalahi ai ka hoʻomaopopo a me ka hoʻoponopono ʻana i nā pilikia geometric paʻakikī.
