Chitsanzo cha Funso Lokambirana pa Circular Arcs
Mu geometry, bwalo ndi chithunzi chozungulira chokhala ndi mfundo zambiri zosangalatsa zophunzirira, chimodzi mwa izo ndi arc. Arc ndi gawo la m'mphepete mwa bwalo lomwe lili pakati pa mfundo ziwiri pa bwalo. M'nkhaniyi, tifufuza zitsanzo zosiyanasiyana za mavuto ndi mayankho awo okhudzana ndi arcs.
Kumvetsetsa Koyambira kwa Ma Circular Arcs
Musanapite ku mafunso achitsanzo, ndikofunikira kumvetsetsa mfundo zina zoyambira kaye:
1. Bwalo:
Bwalo ndi gulu la mfundo zonse zomwe zili pamtunda womwewo zomwe zili kutali ndi malo enaake apakati.
2. Ulalo (Zala):
Utali wa radius ndi mtunda wochokera pakati pa bwalo mpaka pamalo aliwonse m'mphepete mwa bwalo.
3. M'mimba mwake:
Dayamita ndi mtunda wautali kwambiri kuchokera pa mfundo imodzi m'mphepete mwa bwalo kupita ku mfundo ina kumbali inayo kudzera pakati. Dayamita yake ndi kawiri kuposa radius.
4. Uta:
Arc ndi gawo la m'mphepete mwa bwalo. Ngati mfundo A ndi B zili m'mphepete mwa bwalo, ndiye kuti arc AB ndi gawo la bwalo pakati pa A ndi B.
Mafomula Ogwirizana ndi Ma Arcs of Circles
Kuti tiyese kutalika kwa arc, tifunika kumvetsetsa njira zingapo:
1. Utali wa Arc (L):
Kutalika kwa mzere wozungulira ndi kutalika kwa m'mphepete mwa bwalo lomwe limaphatikizapo mzerewo. Fomula yake ndi iyi:
\[
L = r \times \theta
\]
kumene \( r \) ndi radius ya bwalo ndipo \( \theta \) ndi ngodya yapakati mu ma radians yomwe imadutsa arc.
2. Kutalika kwa Arc mu Degrees:
Ngati ngodya yapakati ili mu madigiri, timagwiritsa ntchito fomula iyi:
\[
L = \frac{\theta}{360^\circ} \times 2\pir r
\]
kumene \( \theta \) ndi ngodya yapakati mu madigiri.
Mafunso ndi Kukambirana Zitsanzo
Funso 1: Kuwerengera Utali wa Arc
Funso:
Bwalo lozungulira lili ndi utali wa masentimita 10. Werengani kutalika kwa arc yomwe ili pansi pa ngodya yapakati ya madigiri 60.
Kukambirana:
– Utali wozungulira (\( r \)) = 10 cm
– Ngodya yapakati (\( \theta \)) = 60°
Kugwiritsa ntchito fomula ya kutalika kwa arc mu madigiri:
\[
L = \frac{\theta}{360^\circ} \times 2 \pi r
\]
\[
L = \frac{60^\circ}{360^\circ} \times 2 \pi \times 10 \text{ cm}
\]
\[
L = \frac{1}{6} \times 2 \pi \times 10 \text{ cm}
\]
\[
L = \frac{20\pi}{6} \text{ cm}
\]
\[
L \pafupifupi 10.47 \malemba{cm}
\]
Kotero, kutalika kwa arc ndi pafupifupi 10.47 cm.
Funso 2: Kudziwa Ngodya Yapakati Kuchokera Kutali kwa Arc
Funso:
Popeza bwalo lozungulira la 14 cm lili ndi kutalika kwa arc ya 22 cm. Dziwani ngodya yapakati yomwe imadutsa arc mu madigiri.
Kukambirana:
– Utali wozungulira (\( r \)) = 14 cm
– Kutalika kwa arc (\( L \)) = 22 cm
Gwiritsani ntchito fomula ya kutalika kwa arc kuti mupeze \( \theta \):
\[
L = \frac{\theta}{360^\circ} \times 2 \pi r
\]
M'malo mwa mfundo zodziwika bwino:
\[
22 = \frac{\theta}{360^\circ} \nthawi 2 \pi \nthawi 14
\]
Kudzipatula \( \theta \):
\[
22 = \frac{\theta}{360^\circ} \times 28 \pi
\]
\[
22 \nthawi 360^\circ = \theta \nthawi 28 \pi
\]
\[
7920 = \theta \nthawi 28 \pi
\]
\[
\theta = \frac{7920}{28 \pi}
\]
\[
\theta \pafupifupi 90.72^\circ
\]
Kotero, kukula kwa ngodya yapakati ndi pafupifupi madigiri 90.72.
Funso 3: Kuwerengera Dera la Gawo
Funso:
Gawo la bwalo limapangidwa ndi ngodya yapakati ya madigiri 120 ndi utali wa masentimita 7. Dziwani dera la gawolo.
Kukambirana:
– Utali wozungulira (\( r \)) = 7 cm
– Ngodya yapakati (\( \theta \)) = 120°
Gwiritsani ntchito njira ya gawo la gawo:
\[
A = \frac{\theta}{360^\circ} \times \pi r^2
\]
M'malo mwa mfundo zodziwika bwino:
\[
A = \frac{120^\circ}{360^\circ} \times \pi \times 7^2
\]
\[
A = \frac{1}{3} \times \pi \times 49
\]
\[
A = \frac{49\pi}{3}
\]
\[
A \pafupifupi 51.43 \text{ cm}^2
\]
Kotero, dera la gawoli ndi pafupifupi 51.43 cm².
Funso 4: Kudziwa Arc kuchokera ku Gawo la Gawo
Funso:
Dera la gawo lozungulira lokhala ndi utali wa masentimita 6 ndi 18π cm². Kodi kutalika kwa arc ya gawolo ndi kotani?
Kukambirana:
– Utali wozungulira (\( r \)) = 6 cm
– Malo a gawo (\( A \)) = 18π cm²
Gwiritsani ntchito fomula ya dera la gawo kuti mupeze ngodya yapakati \( \theta \):
\[
A = \frac{\theta}{360^\circ} \times \pi r^2
\]
M'malo mwa mfundo zodziwika bwino:
\[
18\pi = \frac{\theta}{360^\circ} \times \pi \times 6^2
\]
\[
18\pi = \frac{\theta}{360^\circ} \times 36\pi
\]
\[
18 = \frac{\theta}{360^\circ} \nthawi 36
\]
\[
18 \nthawi 360^\circ = \theta \nthawi 36
\]
\[
6480 = \theta \nthawi 36
\]
\[
\theta = \frac{6480}{36}
\]
\[
\theta = 180^\circ
\]
Tsopano, ndi ngodya yapakati ya madigiri 180, timazindikira kutalika kwa arc:
\[
L = \frac{\theta}{360^\circ} \times 2 \pi r
\]
\[
L = \frac{180^\circ}{360^\circ} \times 2 \pi \times 6
\]
\[
L = \frac{1}{2} \nthawi 2 \pi \nthawi 6
\]
\[
L = \pi \ nthawi 6
\]
\[
L \pafupifupi 18.85 \malemba{cm}
\]
Kotero, kutalika kwa uta ndi pafupifupi 18.85 cm.
Mapeto
Kumvetsetsa ma arc ozungulira ndi momwe angawawerengere ndi maziko ofunikira kwambiri pa geometry ndi masamu. Kudzera mu zitsanzo zomwe zafotokozedwa m'nkhaniyi, owerenga akuyembekezeka kumvetsetsa bwino momwe angawerengere kutalika kwa arc, dera la gawo, ndikupeza ngodya yapakati yomwe imadutsa arc. Kumvetsetsa bwino mfundo zazikuluzikulu izi kudzakuthandizani kwambiri kuthetsa mavuto osiyanasiyana okhudzana ndi mabwalo.
