Mehlala ea Lipotso tse Buisanang ka Kamano Pakeng tsa Bolelele ba Arc le Sebaka sa Sector
Lithutong tsa jeometri, haholo-holo thutong ea lisekoere, hangata re kopana le mehopolo ea bolelele ba arc le sebaka sa sector. Mehopolo ena e 'meli e bohlokoa bakeng sa ho utloisisa liketsahalo tse fapaneng tsa jeometri tse amanang le lisekoere. A re qaleng ka ho hlalosa mehopolo ena e 'meli pele re fana ka mehlala ea mathata le litharollo tsa 'ona.
Bolelele ba Arc
Bolelele ba arc ke sebaka se pakeng tsa arc pakeng tsa lintlha tse peli selikalikoeng. Ho bala bolelele ba arc ba selikalikoe, hangata re hloka radius ea selikalikoe (r) le sekhutlo se bohareng (θ) seo arc e kenellang ho sona ka li-radian. Foromo ea ho bala bolelele ba arc (s) e ka ngoloa ka tsela e latelang:
\[ s = r \times \theta \]
Haeba sekhutlo se bohareng se fanoe ka di-degree, re tlameha ho se fetolela ho radians ka:
\[ \theta_{radian} = \theta_{degree} \times \frac{\pi}{180} \]
Sebaka sa Lekala
Lekala ke karolo ea selikalikoe se moeling oa li-radii tse peli le arc pakeng tsa tsona. Ho bala sebaka sa lekala, re sebelisa radius ea selikalikoe (r) le sekhutlo se bohareng (θ). Foromo ea ho bala sebaka sa lekala (A) ke:
\[ A = \frac{1}{2} r^2 \times \theta \]
Jwalo ka bolelele ba arc, haeba sekhutlo se bohareng se lekanngwa ka di-degree, re tlameha ho se fetolela ho radians pele.
Lipotso tsa Mehlala le Puisano
Ho hlakisa mohopolo oa bolelele ba arc le sebaka sa lekala, ha re hlahlobeng mehlala e latelang ea lipotso le lipuisano tsa tsona.
Potso ea 1:
Ha o fuoa selikalikoe se nang le radius ea 10 cm le sekhutlo se bohareng sa likhato tse 60, bala bolelele ba arc le sebaka sa lekala se entsoeng ke sekhutlo.
Puisano:
1. Ho Bala Bolelele ba Arc:
– Taba ea pele, re fetola angle ho tloha ho likhato ho ea ho radians:
\[ \theta = 60 \times \frac{\pi}{180} = \frac{\pi}{3} \, \text{radian} \]
– Ho sebedisa foromo ya bolelele ba arc:
\[ s = r \times \theta \]
\[ s = 10 \makgetlo \frac{\pi}{3} \]
\[ s = \frac{10\pi}{3} \, \text{cm} \]
2. Ho Bala Sebaka sa Lekala:
– Ho sebedisa foromo bakeng sa sebaka sa lekala:
\[ A = \frac{1}{2} r^2 \times \theta \]
\[ A = \frac{1}{2} \times 10^2 \times \frac{\pi}{3} \]
\[ A = \frac{1}{2} \times 100 \times \frac{\pi}{3} \]
\[ A = \frac{100\pi}{6} \]
\[ A = \frac{50\pi}{3} \, \text{cm}^2 \]
Kahoo, bolelele ba arc ke \(\frac{10\pi}{3}\) cm, mme sebaka sa lekala ke \(\frac{50\pi}{3}\) cm².
Potso ea 2:
Selikalikoe se na le radius ea 7 cm le sekhutlo se bohareng se ka tlas'a arc ea radians tse 2. Fumana bolelele ba arc le sebaka sa karolo ea selikalikoe.
Puisano:
1. Ho Bala Bolelele ba Arc:
– Sekhutlo se bohareng se se se le ka har'a radians, kahoo re ka sebelisa foromo ea bolelele ba arc ka kotloloho:
\[ s = r \times \theta \]
\[s = 7 \makhetlo a 2 \]
\[s = 14 \, \mongolo{cm} \]
2. Ho Bala Sebaka sa Lekala:
– Ho sebedisa foromo bakeng sa sebaka sa lekala:
\[ A = \frac{1}{2} r^2 \times \theta \]
\[ A = \frac{1}{2} \makgetlo a 7^2 \makgetlo a 2 \]
\[ A = \frac{1}{2} \makgetlo a 49 \makgetlo a 2 \]
\[ A = 49 \, \mongolo{cm}^2 \]
Kahoo, bolelele ba arc ke 14 cm, 'me sebaka sa lekala ke 49 cm².
Potso ea 3:
Selikalikoe se nang le radius ea 12 cm se na le karolo eo bolelele ba eona ba arc e leng 15\(\pi\) cm. Fumana sekhutlo se bohareng ka likhato le sebaka sa karolo.
Puisano:
1. Ho Fumana Sekhutlo se Bohareng:
– Ho sebedisa foromo ya bolelele ba arc ho fumana sekhutlo se bohareng:
\[ s = r \times \theta \]
\[ 15\pi = 12 \makgetlo \theta \]
\[ \theta = \frac{15\pi}{12} \]
\[ \theta = \frac{5\pi}{4} \, \text{radian} \]
– Fetolela sekhutlo se bohareng ho ba likhato:
\[ \theta = \frac{5\pi}{4} \times \frac{180}{\pi} \]
\[ \theta = \frac{5 \times 180}{4} \]
\[ \theta = 225 \, \mongolo{di-degrees} \]
2. Ho Bala Sebaka sa Lekala:
– Ho sebedisa foromo bakeng sa sebaka sa lekala:
\[ A = \frac{1}{2} r^2 \times \theta \]
\[ A = \frac{1}{2} \times 12^2 \times \frac{5\pi}{4} \]
\[ A = \frac{1}{2} \times 144 \times \frac{5\pi}{4} \]
\[ A = 72 \times \frac{5\pi}{4} \]
\[ A = 90\pi \, \mongolo{cm}^2 \]
Kahoo, sekhutlo se bohareng sa lekala ke di-degree tse 225, mme sebaka sa lekala ke 90\(\pi\) cm².
Qetello
Ho utloisisa kamano pakeng tsa bolelele ba arc le sebaka sa lekala ho hloka kutloisiso e batsi ea melao-motheo ea motheo ea li-circles le tšebeliso e nepahetseng ea liforomo. Ka mathata a mokhoa a kaholimo, re ka bona bohlokoa ba ho tseba phetoho ea likhutlo le ho sebelisa liforomo ka kotloloho moelelong oa jeometri e chitja. Mohato o mong le o mong puisanong ea bothata o re thusa ho utloisisa hore na liforomo li sebetsa joang le hore na re ka li sebelisa joang ka katleho.
Ka ho tswela pele ho itloaetsa le ho utlwisisa metheo e hlalositsweng, re tla ba le bokgoni bo eketsehileng ba ho rarolla mathata a amanang le bolelele ba arc le sebaka sa lekala, mme sena se tla ba molemo haholo ditshebedisong tse fapaneng tsa dipalo le tsa mahlale.
