Uchburchak yuzi

Ushbu darsda uchburchak yuzini turli ma’lumotlardan kelib chiqib hisoblashning bir necha usuli haqida so‘z yuritamiz. Bitta shakl — bir necha formula.

§1. Asos va balandlik orqali

Eng oddiy va eng mashhur usul.

Qoida. Uchburchak yuzi asos bilan unga tushirilgan balandlik ko‘paytmasining yarmiga teng:

$$S = \frac{1}{2} a h_a$$

Nega yarmi? Chunki uchburchak — to‘g‘ri to‘rtburchakning aynan yarmidir: uni diagonal bo‘yicha kessak, ikkita teng uchburchak chiqadi.

Misol. Asosi $a = 12$ sm, balandligi $h = 5$ sm. Yuzini toping.
Yechish. Formulaga qo‘yamiz:

$$S = \frac{1}{2} \cdot 12 \cdot 5 = \frac{60}{2} = 30$$

Javob. $S = 30$ sm².

§2. Ikki tomon va burchak orqali

Balandlikni bilmasak ham, ikki tomon va ular orasidagi burchak yetarli.

Qoida. $$ S = \frac{1}{2} ab \sin C $$

Misol. $a = 6$, $b = 8$, ular orasidagi burchak $C = 30°$ ($\sin 30° = 0{,}5$). Yuzini toping.
Yechish.

$$S = \frac{1}{2} \cdot 6 \cdot 8 \cdot 0{,}5 = \frac{48 \cdot 0{,}5}{2} = \frac{24}{2} = 12$$

Javob. $S = 12$.

Savol. Burchak $90°$ bo‘lsa nima bo‘ladi? $\sin 90° = 1$, formula $S = \tfrac{1}{2}ab$ ga aylanadi — bu ikki katetli to‘g‘ri burchakli uchburchak yuzi!

§3. Uch tomon orqali — Geron formulasi

Faqat uch tomon berilsa, Geron formulasidan foydalanamiz.

Qoida. Yarim perimetr $p = \dfrac{a+b+c}{2}$ bo‘lsa,

$$S = \sqrt{p(p-a)(p-b)(p-c)}$$

Misol. Tomonlari $a = 3$, $b = 4$, $c = 5$. Yuzini toping.
Yechish. Avval yarim perimetrni topamiz:

$$p = \frac{3+4+5}{2} = \frac{12}{2} = 6$$

$$S = \sqrt{6(6-3)(6-4)(6-5)} = \sqrt{6 \cdot 3 \cdot 2 \cdot 1} = \sqrt{36} = 6$$

Javob. $S = 6$.

Eslatma. Bir masalada bir necha usul mos kelishi mumkin. Berilgan ma’lumotga qarang: balandlik bormi — birinchi formula; ikki tomon va burchakmi — ikkinchi; faqat tomonlarmi — Geron.

Misol. Teng tomonli uchburchak tomoni $a = 4$. Yuzini toping.
Yechish. Burchak $60°$, $\sin 60° = \tfrac{\sqrt{3}}{2}$:

$$S = \frac{1}{2} \cdot 4 \cdot 4 \cdot \frac{\sqrt{3}}{2} = \frac{16\sqrt{3}}{4} = 4\sqrt{3} \approx 6{,}93$$

Javob. $S = 4\sqrt{3} \approx 6{,}93$.

§ Lug‘at

yuz — area
asos — base
balandlik — height (altitude)
yarim perimetr — semi-perimeter
Geron formulasi — Heron's formula
ko‘paytma — product
diagonal — diagonal
teng tomonli — equilateral