Isibonelo Sombuzo Wengxoxo Nge-Circular Arcs
Ku-geometry, indilinga iyisithombe esibanzi esinemiqondo eminingi ethakazelisayo yokufunda, enye yayo i-arc. I-arc yingxenye yomphetho wendilinga ephakathi kwamaphuzu amabili endilinga. Kulesi sihloko, sizohlola izinkinga ezahlukahlukene zezibonelo kanye nezixazululo zazo ezihlobene nama-arc.
Ukuqonda Okuyisisekelo Kwe-Circular Arcs
Ngaphambi kokuqhubekela emibuzweni yesibonelo, kubalulekile ukuqonda imiqondo eyisisekelo kuqala:
1. Isiyingi:
Indilinga iqoqo lawo wonke amaphuzu endizeni aqhelelene kakhulu nephuzu eliphakathi nendawo elinikeziwe.
2. Irediyasi (Iminwe):
Irediyasi ibanga elisuka enkabeni yendilinga kuya kunoma iyiphi indawo esemaphethelweni endilinga.
3. Ububanzi:
Ububanzi buyibanga elide kakhulu ukusuka endaweni eyodwa emaphethelweni esiyingi kuya kwenye indawo ngakolunye uhlangothi phakathi nendawo. Ububanzi buphindwe kabili kune-radius.
4. Ukukhothama:
I-arc iyingxenye yomphetho wesiyingi. Uma amaphuzu u-A no-B elele onqenqemeni lwesiyingi, khona-ke i-arc u-AB iyingxenye yesiyingi esiphakathi kuka-A no-B.
Amafomula Ahlobene Nama-Arcs Emibuthano
Ukuze silinganise ubude be-arc, sidinga ukuqonda amafomula amaningana:
1. Ubude be-Arc (L):
Ubude be-arc eyindilinga ubude bomphetho wendilinga ohlanganisa i-arc. Ifomula ithi:
\[
L = r \izikhathi \theta
\]
lapho \( r \) kuyirediyasi yesiyingi kanye \( \theta \) kuyi-engeli ephakathi kuma-radians ehlangana ne-arc.
2. Ubude be-Arc ngama-degree:
Uma i-engeli ephakathi ingamadigri, sisebenzisa ifomula:
\[
L = \frac{\theta}{360^\circ} \izikhathi 2\pir r
\]
lapho \( \theta \) kuyi-engeli ephakathi ngamadigri.
Imibuzo Eyisibonelo Nengxoxo
Umbuzo 1: Ukubala Ubude be-Arc
Umbuzo:
Indilinga inobubanzi obuyi-10 cm. Bala ubude be-arc obungaphansi kwe-engeli ephakathi yama-degrees angu-60.
Ingxoxo:
– Irediyasi yesiyingi (\( r \)) = 10 cm
– I-engeli ephakathi (\( \theta \)) = 60°
Ukusebenzisa ifomula yobude be-arc ngamadigri:
\[
L = \frac{\theta}{360^\circ} \izikhathi 2 \pi r
\]
\[
L = \frac{60^\circ}{360^\circ} \times 2 \pi \times 10 \text{ cm}
\]
\[
L = \frac{1}{6} \times 2 \pi \times 10 \text{ cm}
\]
\[
L = \frac{20\pi}{6} \text{ cm}
\]
\[
L \cishe 10.47 \umbhalo{cm}
\]
Ngakho-ke, ubude be-arc bungaba ngu-10.47 cm.
Umbuzo 2: Ukunquma i-Engela Ephakathi Kusukela Kubude Be-Arc
Umbuzo:
Uma ubheka indilinga enobubanzi obungu-14 cm, ubude bayo bungu-22 cm. Thola i-engeli ephakathi ehlangana ne-arc ngamadigri.
Ingxoxo:
– Irediyasi yesiyingi (\( r \)) = 14 cm
– Ubude be-arc (\( L \)) = 22 cm
Sebenzisa ifomula yobude be-arc ukuthola \( \theta \):
\[
L = \frac{\theta}{360^\circ} \izikhathi 2 \pi r
\]
Faka amanani aziwayo esikhundleni sawo:
\[
22 = \frac{\theta}{360^\circ} \izikhathi 2 \pi \izikhathi 14
\]
Ukuhlukaniswa \( \theta \):
\[
22 = \frac{\theta}{360^\circ} \izikhathi ezingu-28 \pi
\]
\[
22 \izikhathi 360^\circ = \theta \izikhathi 28 \pi
\]
\[
7920 = \theta \izikhathi ezingu-28 \pi
\]
\[
\theta = \frac{7920}{28 \pi}
\]
\[
\theta \cishe 90.72^\circ
\]
Ngakho-ke, ubukhulu be-engeli ephakathi bungama-degrees angu-90.72.
Umbuzo 3: Ukubala Indawo Yomkhakha
Umbuzo:
Umkhakha wendilinga wakhiwa nge-engeli ephakathi yama-degrees angu-120 enobubanzi obungu-7 cm. Nquma indawo yomkhakha.
Ingxoxo:
– Irediyasi yesiyingi (\( r \)) = 7 cm
– I-engeli ephakathi (\( \theta \)) = 120°
Sebenzisa ifomula yendawo yomkhakha:
\[
A = \frac{\theta}{360^\circ} \times \pi r^2
\]
Faka amanani aziwayo esikhundleni sawo:
\[
A = \frac{120^\circ}{360^\circ} \times \pi \times 7^2
\]
\[
A = \frac{1}{3} \izikhathi \pi \izikhathi 49
\]
\[
A = \frac{49\pi}{3}
\]
\[
A \cishe 51.43 \umbhalo{cm}^2
\]
Ngakho-ke, indawo yomkhakha icishe ibe ngu-51.43 cm².
Umbuzo 4: Ukunquma i-Arc kusuka endaweni yomkhakha
Umbuzo:
Indawo yesigaba esiyindilinga esinobubanzi obungu-6 cm ingu-18π cm². Bungakanani ubude be-arc yalesi sigaba?
Ingxoxo:
– Irediyasi yesiyingi (\( r \)) = 6 cm
– Indawo yomkhakha (\( A \)) = 18π cm²
Sebenzisa ifomula yendawo yesigaba ukuthola i-engeli ephakathi \( \theta \):
\[
A = \frac{\theta}{360^\circ} \times \pi r^2
\]
Faka amanani aziwayo esikhundleni sawo:
\[
18\pi = \frac{\theta}{360^\circ} \times \pi \times 6^2
\]
\[
18\pi = \frac{\theta}{360^\circ} \times 36\pi
\]
\[
18 = \frac{\theta}{360^\circ} \izikhathi ezingu-36
\]
\[
18 \izikhathi 360^\circ = \theta \izikhathi 36
\]
\[
6480 = \theta \izikhathi 36
\]
\[
\theta = \frac{6480}{36}
\]
\[
\theta = 180^\circ
\]
Manje, nge-engeli ephakathi yama-degrees angu-180, sinquma ubude be-arc:
\[
L = \frac{\theta}{360^\circ} \izikhathi 2 \pi r
\]
\[
L = \frac{180^\circ}{360^\circ} \izikhathi 2 \pi \izikhathi 6
\]
\[
L = \frac{1}{2} \izikhathi 2 \pi \izikhathi 6
\]
\[
L = \pi \izikhathi 6
\]
\[
L \cishe 18.85 \umbhalo{cm}
\]
Ngakho-ke, ubude bomnsalo bungaba ngu-18.85 cm.
Isiphetho
Ukuqonda ama-arc ayindilinga nokuthi ungawabala kanjani kuyisisekelo esibalulekile ku-geometry kanye nezibalo ngokujwayelekile. Ngezibonelo okuxoxwe ngazo kulesi sihloko, abafundi kulindeleke ukuthi bathole ukuqonda okungcono kokuthi bangabala kanjani ubude be-arc, indawo yesigaba, futhi banqume i-engeli ephakathi ehlangana ne-arc. Ukuqonda kahle le mibono eyisisekelo kuzosiza kakhulu ekuxazululeni izinkinga ezahlukahlukene ezihilela imibuthano.
