Ichki va tashqi chizilgan aylana

Bu nima va nega kerak

Muhandislik va mexanizm detallari tayyorlashda ko'pburchak shaklidagi qismni aylanuvchi valga o'rnatish (tashqi aylana) yoki uning markaziga dumaloq teshik ochish (ichki aylana) ko'p uchraydi.

Geometriyada har qanday shaklga ham aylana chizib bo'lavermaydi:

• Ixtiyoriy uchburchakka ichki ham, tashqi ham aylana chizish mumkin.
• To'rtburchaklarda esa faqat ma'lum qat'iy shartlar bajarilgandagina aylana chiziladi.

Ushbu darsda ko'pburchaklarga ichki va tashqi chizilgan aylanalar markazlari, radius formulalari va ularning yuzalar bilan bog'liqligini o'rganamiz.

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Asosiy g'oya va ta'riflar

       Tashqi aylana                       Ichki aylana
           /\                                  /\
          /  \                                /  \
         / O  \ (uchlar aylanada)            / O  \ (tomonlarga urinadi)
        /______\                            /______\

1. Uchburchakka tashqi chizilgan aylana

[!IMPORTANT] Ta'rif. Aylananing barcha uchlari aylanada yotsa, bu aylana uchburchakka tashqi chizilgan aylana deyiladi.

• Markazi: Tomonlarning o'rta perpendikulyarlari kesishgan nuqtada.
• Asosiy formula: $R = \frac{abc}{4S}$
• To'g'ri burchakli uchburchakda: Markaz gipotenuzaning o'rtasida yotadi: $R = \frac{c}{2}$
• Muntazam uchburchakda: $R = \frac{a}{\sqrt{3}} = \frac{a\sqrt{3}}{3}$

2. Uchburchakka ichki chizilgan aylana

[!IMPORTANT] Ta'rif. Aylana uchburchakning barcha tomonlariga urinsa, u ichki chizilgan aylana deyiladi.

• Markazi: Ichki burchaklar bissektrisalari kesishgan nuqtada.
• Asosiy formula: $r = \frac{S}{p}$ bu yerda $p = \frac{a + b + c}{2}$ — yarim perimetr, $S$ — yuza.
• To'g'ri burchakli uchburchakda: $r = \frac{a + b - c}{2} = p - c$
• Muntazam uchburchakda: $r = \frac{a\sqrt{3}}{6} = \frac{R}{2}$

3. To'rtburchaklar va aylanalar

[!IMPORTANT] Aylanaga ichki chizilgan to'rtburchak. To'rtburchakka tashqi aylana chizish mumkin bo'lishi uchun qarama-qarshi burchaklar yig'indisi $180^\circ$ bo'lishi zarur va yetarli: $\angle A + \angle C = \angle B + \angle D = 180^\circ$ (Misollar: to'g'ri to'rtburchak, kvadrat, teng yonli trapetsiya).

[!IMPORTANT] Aylanaga tashqi chizilgan to'rtburchak. To'rtburchakka ichki aylana chizish mumkin bo'lishi uchun qarama-qarshi tomonlar yig'indilari teng bo'lishi zarur va yetarli: $a + c = b + d$ Yuzi: $S = p \cdot r$. (Misollar: romb, kvadrat, tomonlari shartni qanoatlantiruvchi trapetsiya).

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Yechilgan misollar

1-misol. To'g'ri burchakli uchburchakda $R$ va $r$

Katetlari $6\text{ sm}$ va $8\text{ sm}$ bo'lgan to'g'ri burchakli uchburchakka tashqi va ichki aylanalar radiuslarini toping.

• Gipotenuza: $c = \sqrt{6^2 + 8^2} = 10\text{ sm}$.
• Tashqi aylana radiusi: $R = \frac{c}{2} = \frac{10}{2} = 5\text{ sm}$.
• Ichki aylana radiusi: $r = \frac{6 + 8 - 10}{2} = 2\text{ sm}$.

Javob: $R = 5\text{ sm}$, $r = 2\text{ sm}$.

2-misol. Muntazam uchburchakda radiuslar

Tomoni $12\text{ sm}$ bo'lgan muntazam uchburchakka tashqi va ichki aylanalar radiuslarini toping.

• Tashqi aylana radiusi: $R = \frac{a}{\sqrt{3}} = \frac{12}{\sqrt{3}} = 4\sqrt{3}\text{ sm}$.
• Ichki aylana radiusi: $r = \frac{R}{2} = \frac{4\sqrt{3}}{2} = 2\sqrt{3}\text{ sm}$.

Javob: $R = 4\sqrt{3}\text{ sm}$, $r = 2\sqrt{3}\text{ sm}$.

3-misol. Teng yonli uchburchakda $R$ va $r$

Asosi $12\text{ sm}$, yon tomonlari $10\text{ sm}$ bo'lgan teng yonli uchburchakka ichki va tashqi aylanalar radiuslarini hisoblang.

• Asosga tushirilgan balandlik: $h = \sqrt{10^2 - 6^2} = 8\text{ sm}$.
• Yuza: $S = \frac{1}{2} \cdot 12 \cdot 8 = 48\text{ sm}^2$.
• Yarim perimetr: $p = \frac{10 + 10 + 12}{2} = 16\text{ sm}$.
• Ichki aylana radiusi: $r = \frac{S}{p} = \frac{48}{16} = 3\text{ sm}$.
• Tashqi aylana radiusi: $R = \frac{abc}{4S} = \frac{10 \cdot 10 \cdot 12}{4 \cdot 48} = \frac{1200}{192} = 6{,}25\text{ sm}$.

Javob: $r = 3\text{ sm}$, $R = 6{,}25\text{ sm}$.

4-misol. Aylanaga tashqi chizilgan trapetsiya

Aylanaga tashqi chizilgan teng yonli trapetsiyaning asoslari $4\text{ sm}$ va $16\text{ sm}$. Uning yon tomonini va aylananing radiusini toping.

• Qarama-qarshi tomonlar yig'indisi: $a + b = 2c \implies 4 + 16 = 2c \implies c = 10\text{ sm}$.
• Balandlik: $h = \sqrt{10^2 - ((16 - 4)/2)^2} = \sqrt{100 - 36} = 8\text{ sm}$.
• Aylana radiusi: $r = \frac{h}{2} = \frac{8}{2} = 4\text{ sm}$.

Javob: Yon tomoni $10\text{ sm}$, radiusi $4\text{ sm}$.

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Mashq

1. Katetlari $8\text{ sm}$ va $15\text{ sm}$ bo'lgan to'g'ri burchakli uchburchakka tashqi va ichki aylanalar radiuslarini toping.
2. Tomoni $6\text{ sm}$ bo'lgan muntazam uchburchakka ichki va tashqi aylanalar radiuslarini hisoblang.
3. Yon tomonlari $5\text{ sm}$, asosi $6\text{ sm}$ bo'lgan teng yonli uchburchakka ichki chizilgan aylana radiusini toping.
4. Aylanaga ichki chizilgan $ABCD$ to'rtburchakda $\angle A : \angle B : \angle C = 2 : 3 : 4$. Uning barcha burchaklarini toping.
5. Aylanaga tashqi chizilgan to'rtburchakning perimetri $48\text{ sm}$. Qarama-qarshi ikki tomoni yig'indisini toping.
6. Gipotenuzasi $20\text{ sm}$ bo'lgan to'g'ri burchakli uchburchakka tashqi chizilgan aylana radiusini toping.
7. Rombning tomoni $10\text{ sm}$, o'tkir burchagi $30^\circ$. Unga ichki chizilgan aylana radiusini toping.
8. Aylanaga tashqi chizilgan teng yonli trapetsiyaning yon tomoni $8\text{ sm}$. Uning perimetrini va o'rta chizig'ini hisoblang.
9. Uchburchak tomoni $10\text{ sm}$, uning qarshisidagi burchak $30^\circ$. Tashqi chizilgan aylana radiusini toping.
10. Tomonlari $12\text{ sm}$ va $16\text{ sm}$ bo'lgan to'g'ri to'rtburchakka tashqi chizilgan aylana radiusini toping.

Javoblar va yechim yo'llari

1. Gipotenuza $c = 17\text{ sm}$. $R = 17 : 2 = 8{,}5\text{ sm}$. $r = \frac{8 + 15 - 17}{2} = 3\text{ sm}$. Javob: $R = 8{,}5\text{ sm}$, $r = 3\text{ sm}$.
2. $R = \frac{6}{\sqrt{3}} = 2\sqrt{3}\text{ sm}$, $r = \frac{R}{2} = \sqrt{3}\text{ sm}$. Javob: $R = 2\sqrt{3}\text{ sm}$, $r = \sqrt{3}\text{ sm}$.
3. $h = \sqrt{5^2 - 3^2} = 4\text{ sm}$. $S = 12\text{ sm}^2$, $p = 8\text{ sm}$. $r = 12 : 8 = 1{,}5\text{ sm}$. Javob: $1{,}5\text{ sm}$.
4. $\angle A + \angle C = 180^\circ \implies 6x = 180^\circ \implies x = 30^\circ$. Burchaklar: $60^\circ, 90^\circ, 120^\circ, 90^\circ$. Javob: $60^\circ, 90^\circ, 120^\circ, 90^\circ$.
5. $a + c = P / 2 = 48 / 2 = 24\text{ sm}$. Javob: $24\text{ sm}$.
6. $R = c / 2 = 20 / 2 = 10\text{ sm}$. Javob: $10\text{ sm}$.
7. $h = 10 \cdot \sin 30^\circ = 5\text{ sm}$. $r = h / 2 = 2{,}5\text{ sm}$. Javob: $2{,}5\text{ sm}$.
8. $a + b = 2c = 16\text{ sm}$. $P = 32\text{ sm}$. O'rta chiziq $m = 16 / 2 = 8\text{ sm}$. Javob: $P = 32\text{ sm}$, $m = 8\text{ sm}$.
9. $2R = \frac{10}{\sin 30^\circ} = 20 \implies R = 10\text{ sm}$. Javob: $10\text{ sm}$.
10. $d = \sqrt{12^2 + 16^2} = 20\text{ sm}$. $R = d / 2 = 10\text{ sm}$. Javob: $10\text{ sm}$.

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Keng tarqalgan xatolar

1. Ixtiyoriy to'rtburchakka ichki yoki tashqi aylana chizish mumkin deb o'ylash. ✗ Har qanday to'rtburchakka ichki aylana chizishga urinish. ✓ Ichki aylana faqat $a + c = b + d$ bo'lganda, tashqi aylana esa $\angle A + \angle C = 180^\circ$ bo'lgandagina mavjud bo'ladi.

2. $R = c/2$ formulasini barcha uchburchaklarga qo'llash. ✗ O'tkir yoki o'tmas burchakli uchburchakda $c/2$ ni tashqi radius deb olish. ✓ $R = c/2$ faqat va faqat to'g'ri burchakli uchburchak uchungina o'rinli.

3. Ichki aylana radiusi formulasida $p$ o'rniga $P$ ni qo'yish. ✗ $r = \frac{S}{P}$ deb hisoblash. ✓ Formula: $r = \frac{S}{p}$, bu yerda $p$ — yarim perimetr ($p = P / 2$).

4. Ichki va tashqi chizilgan to'rtburchaklar alomatlarini adashtirish. ✗ Ichki chizilganda tomonlar yig'indisini, tashqi chizilganda burchaklar yig'indisini tenglashtirish. ✓ Ichki chizilganda — qarama-qarshi burchaklar yig'indisi $180^\circ$; tashqi chizilganda — qarama-qarshi tomonlar yig'indisi teng bo'ladi.

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Keyingi qadam

Uchburchak va to'rtburchaklar uchun aylanalar xossalarini umumlashtirdik. Keyingi darsda barcha tomonlari va burchaklari teng bo'lgan simmetrik shakllar — muntazam ko'pburchaklar mavzusiga o'tamiz.