Imibuzo Eyisibonelo Exoxa Ngencazelo Yesiyingi
Indilinga ingenye yezimo eziyisisekelo zejiyometri esivame ukuhlangana nazo ekuphileni kwansuku zonke. Kumathematika, imibuthano inezincazelo nezici ezihlukile. Lesi sihloko sizohlola incazelo yendilinga kanye nezakhi zayo ezihlobene ngokujulile, kanye nokunikeza izibonelo eziningana zezinkinga kanye nezixazululo zokujulisa ukuqonda kwethu imibuthano.
Incazelo ye-Circle
Indilinga iyiqoqo lawo wonke amaphuzu endizeni aqhelelene nephuzu eliqondile elibizwa ngokuthi isikhungo sendilinga. Ibanga eliphakathi kwesikhungo nanoma yiliphi iphuzu elisendilinga libizwa ngokuthi i-radius yendilinga. Isibalo esijwayelekile sendilinga enesikhungo endaweni \((h, k)\) kanye ne-radius \(r\) sinikezwa ngu:
\[ (x – h)^2 + (y – k)^2 = r^2 \]
Lapho:
– \((h, k)\) yizixhumanisi zesikhungo sendilinga,
– \(r\) irediyasi yesiyingi,
– \(x\) kanye \(y\) yizixhumanisi zanoma yiliphi iphuzu elisendilinga.
Izinto Zendilinga
Ngaphambi kokuthi singene emibuzweni eyisibonelo, kungumqondo omuhle ukwazi ezinye zezinto ezibalulekile embuthanweni:
1. Isikhungo Sendilinga: Iphuzu eliqinile eliyisikhungo sawo wonke amaphuzu aqhelelene ngebanga elifanayo.
2. Irediyasi (r): Ibanga elisuka enkabeni yendilinga kuya kunoma iyiphi indawo endilinga.
3. Ububanzi (d): Umugqa oqondile odlula phakathi kwendilinga futhi uxhumanisa amaphuzu amabili endilinga, unobude obuphindwe kabili kunerediyasi (\(d = 2r\)).
4. I-Arc: Ingxenye yomjikelezo wendilinga ephakathi kwamaphuzu amabili endilinga.
5. I-Chord: Umugqa oqondile oxhumanisa amaphuzu amabili esiyingini kodwa ongadluli phakathi nendawo.
6. I-Apothem: Ibanga elifushane kakhulu ukusuka enkabeni yendilinga kuya ku-chord.
7. I-Engela Ephakathi: I-engela eyakhiwe ngama-radii amabili avela enkabeni yendilinga.
8. I-Angle Yesiyingi: I-engeli eyakhiwe ama-chord amabili ahlangana endaweni eyodwa esiyingini.
Imibuzo Nezingxoxo Eziyisibonelo
Isibonelo Umbuzo 1
Umbuzo: Unikezwe indilinga enesikhungo endaweni \((3, 4)\) futhi idlula endaweni \((7, 4)\). Thola isibalo sendilinga.
Ingxoxo:
Ukuze sithole i-equation yendilinga, sidinga ukwazi i-radius yayo kuqala. Njengoba indilinga idlula ephuzwini \((7, 4)\), singabala ibanga eliphakathi kwaleli phuzu kanye nesikhungo sendilinga, okungu-\((3, 4)\).
\[
r = \sqrt{(x_2 – x_1)^2 + (y_2 – y_1)^2}
\]
\[
r = \sqrt{(7 – 3)^2 + (4 – 4)^2}
\]
\[
r = \sqrt{4^2 + 0^2}
\]
\[
r = 4
\]
Njengoba isikhungo sise-\((3, 4)\) kanye ne-radius 4, i-equation yendilinga ithi:
\[
(x – 3)^2 + (y – 4)^2 = 4^2
\]
\[
(x – 3)^2 + (y – 4)^2 = 16
\]
Isibonelo Umbuzo 2
Umbuzo: Thola indawo kanye nomjikelezo wendilinga enobubanzi obungu-5 cm.
Ingxoxo:
– Indawo yendilinga (A) ingabalwa kusetshenziswa ifomula \(A = \pi r^2\),
\[
A = \pi \izikhathi 5^2
\]
\[
A = 25\pi \umbhalo{cm}^2
\]
Uma \(\pi \approx 3.14\), khona-ke:
\[
A \cishe izikhathi ezingu-25 \izikhathi ezingu-3.14 = 78.5 \umbhalo{cm}^2
\]
– Umjikelezo wendilinga (C) ungabalwa kusetshenziswa ifomula \(C = 2\pi r\),
\[
C = 2 \izikhathi \pi \izikhathi 5
\]
\[
C = 10\pi \umbhalo{cm}
\]
Uma \(\pi \approx 3.14\), khona-ke:
\[
C \cishe kube yi-10 \izikhathi 3.14 = 31.4 \umbhalo{cm}
\]
Isibonelo Umbuzo 3
Umbuzo: Indilinga inesikhungo endaweni ethi O kanye nerediyasi engu-7 cm. Uma kudwetshwe i-chord enobude obuyi-10 cm endilinga, nquma ibanga elifushane kakhulu ukusuka enkabeni ethi O kuya ku-chord.
Ingxoxo:
Ukuze sithole ibanga elifushane kakhulu ukusuka enkabeni kuya ku-chord, sisebenzisa umqondo we-apothem, okuyibanga elifushane kakhulu ukusuka enkabeni kuya ku-chord. Njengoba sine-radius engu-7 cm kanye nobude be-chord obungu-10 cm, singabhekana nale nkinga sisebenzisa unxantathu ofanele owakhiwe.
Uma i-midpoint ye-chord iyi-point C, khona-ke i-OC yi-apothem esiyifunayo. Uma u-A no-B beyiziphetho ze-chord, khona-ke i-AC ne-BC ngayinye ingu-5 cm (isigamu se-10 cm).
Kusukela kunxantathu i-OAC ebekwe ngakwesokudla ku-C, sisebenzisa i-Pythagorean theorem:
\[
OA^2 = OC^2 + AC^2
\]
\[
7^2 = OC^2 + 5^2
\]
\[
49 = OC^2 + 25
\]
\[
OC^2 = 49 – 25
\]
\[
OC^2 = 24
\]
\[
OC = \sqrt{24} = 2\sqrt{6} \umbhalo{cm}
\]
Isibonelo Umbuzo 4
Umbuzo: Unikezwe indilinga ene-equation \(x^2 + y^2 + 6x – 8y + 9 = 0\). Thola isikhungo kanye nerediyasi yendilinga.
Ingxoxo:
Ukuze sithole isikhungo kanye nerediyasi, siguqula i-equation ibe yifomu ejwayelekile:
\[
x^2 + y^2 + 6x – 8y + 9 = 0
\]
Sixazulula ngokuthuthukisa ifomu le-quadratic:
\[
x^2 + 6x + y^2 – 8y = -9
\]
Ukwengeza nokususa (6/2)\(^2\) ku-x-term kanye (8/2)\(^2\) ku-y-term:
\[
x^2 + 6x + 9 + y^2 – 8y + 16 = -9 + 9 + 16
\]
\[
(x + 3)^2 + (y – 4)^2 = 16
\]
Kusukela ku-equation \((x + 3)^2 + (y – 4)^2 = 16\), sithola ukuthi isikhungo sendilinga ngu-\((-3, 4)\) kanti irediyasi ingu-\(r = \sqrt{16} = 4\).
Isiphetho
Indilinga ingumqondo oyisisekelo nobalulekile ku-geometry. Ngezibonelo zezinkinga nezingxoxo ezingenhla, singathola ukuqonda okujulile ngezici nezakhiwo ezahlukahlukene zendilinga. Ukuqonda lesi sihloko kuzokwenza kube lula ukuqonda nokuxazulula izinkinga ze-geometry eziyinkimbinkimbi kakhulu.
