16) ABCDEFGH nuntazam sakkizburchak berilgan. Agar ABCD to'rtburchak yuzi 12 ga teng bo'lsa, sakkizburchak yuzini toping. 17) ABCDE beshburchakda $$ \mathrm{AE}=\mathrm{AB} ; \mathrm{BC}=\mathrm{CD} $$ va $$ \angle A=\angle C=90^{\circ} $$ bo'lsin. Agar $$ \mathrm{AC}=6 $$ bo'lsa, bu beshburchak yuzini toping. 18) ABC uchburchakda $$ \angle A C B=100^{\circ}, \angle A B C= $$ $$ 50^{\circ} $$. Markazi $$ C $$ nuqtada bo'lgan, radiusi AC ga teng aylana AB va CB tomonlarni mos ravishda $$ D $$ va $$ E $$ nuqtalarda kesadi. AD va DE to'g'ri chiziqlar orasidagi o'tkir burchakni toping. 19) $$ A B C $$ uchburchak yuzasi 60 ga teng. $$ \overrightarrow{B E}=-\frac{1}{2} \overrightarrow{B A} $$ bo' 'sa, $$ A C E $$ uchburchak yuzini toping.

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Keling, har bir masalani alohida ko'rib chiqamiz va yechimlarni bosqichma-bosqich tushuntiramiz.

### 16) ABCDEFGH nuntazam sakkizburchak

Berilgan shartlar:
- ABCD to'rtburchakning yuzi $$ S_{ABCD} = 12 $$.

Nuntazam sakkizburchakning yuzi to'rtburchakning yuziga teng bo'lgan to'rtburchaklar soniga ko'paytirilganda topiladi. Nuntazam sakkizburchak 8 ta teng to'rtburchakdan iborat.

Shunday qilib, sakkizburchakning yuzi:
$$
S_{ABCDEFGH} = 2 \times S_{ABCD} = 2 \times 12 = 24
$$

### 17) ABCDE beshburchak

Berilgan shartlar:
- $$ AE = AB $$
- $$ BC = CD $$
- $$ \angle A = \angle C = 90^{\circ} $$
- $$ AC = 6 $$

Beshburchakni ikkita to'rtburchakka bo'lishimiz mumkin: $$ ABCD $$ va $$ ADE $$.

1. $$ \triangle ABC $$ to'rtburchakning yuzi:
   - $$ AB = AE = x $$
   - $$ BC = CD = y $$
   - $$ AC = \sqrt{x^2 + y^2} = 6 $$

2. Yuzi:
   $$
   S_{ABCD} = AB \cdot BC = x \cdot y
   $$

3. $$ x^2 + y^2 = 36 $$ (Pifagor teoremasi).

Beshburchakning yuzi:
$$
S_{ABCDE} = S_{ABCD} + S_{ADE} = xy + \frac{1}{2} \cdot x \cdot y = xy + \frac{1}{2}xy = \frac{3}{2}xy
$$

Bu yerda $$ x $$ va $$ y $$ ni aniqlashimiz kerak. Buni yechish uchun qo'shimcha ma'lumot kerak.

### 18) ABC uchburchak

Berilgan shartlar:
- $$ \angle ACB = 100^{\circ} $$
- $$ \angle ABC = 50^{\circ} $$

Bundan $$ \angle BAC = 30^{\circ} $$ bo'ladi.

1. $$ D $$ va $$ E $$ nuqtalarini topamiz:
   - $$ D $$ nuqtasi $$ AB $$ tomonida, $$ E $$ nuqtasi $$ CB $$ tomonida joylashgan.

2. $$ AD $$ va $$ DE $$ to'g'ri chiziqlar orasidagi o'tkir burchakni topish uchun:
   - $$ \angle ADB $$ va $$ \angle CEB $$ burchaklarini hisoblaymiz.

Burchaklar orasidagi munosabatni aniqlash uchun trigonometrik formulalardan foydalanamiz.

### 19) ABC uchburchak

Berilgan shartlar:
- $$ S_{ABC} = 60 $$
- $$ \overrightarrow{BE} = -\frac{1}{2} \overrightarrow{BA} $$

Bu shartlar asosida $$ A $$, $$ B $$, va $$ C $$ nuqtalarini aniqlaymiz. $$ E $$ nuqtasi $$ B $$ nuqtasidan $$ A $$ nuqtasiga qarab harakatlanadi.

1. $$ A $$ va $$ C $$ nuqtalari orasidagi masofani aniqlaymiz.
2. $$ A $$, $$ C $$, va $$ E $$ nuqtalari orasidagi uchburchakning yuzini topamiz.

Buni yechish uchun $$ S_{ACE} $$ ni hisoblaymiz.

Keling, 17, 18 va 19 masalalarini yechish uchun zarur bo'lgan hisob-kitoblarni amalga oshiramiz.

### 17, 18, 19 masalalarini yechish uchun zarur bo'lgan hisob-kitoblar

Keling, 17, 18 va 19 masalalarini yechish uchun zarur bo'lgan hisob-kitoblarni amalga oshiramiz.

#### 17) Beshburchakning yuzi

Beshburchakning yuzi uchun $$ x $$ va $$ y $$ ni aniqlashimiz kerak. Buni yechish uchun $$ x^2 + y^2 = 36 $$ va $$ S_{ABCDE} = \frac{3}{2}xy $$ formulalaridan foydalanamiz.

#### 18) O'tkir burchakni topish

O'tkir burchakni topish uchun trigonometrik formulalardan foydalanamiz.

#### 19) Uchburchakning yuzi

Uchburchakning yuzini topish uchun $$ S_{ACE} $$ ni hisoblaymiz.

Keling, bu masalalarni yechish uchun zarur bo'lgan hisob-kitoblarni amalga oshiramiz.

Buni amalga oshirish uchun zarur bo'lgan funksiyalarni chaqiramiz.

```json
{
  "tool_uses": [
    {
      "recipient_name": "functions.solve_system_equations",
      "parameters": {
        "equations": [
          "x^2 + y^2 = 36",
          "S_{ABCDE} = \frac{3}{2}xy"
        ]
      }
    },
    {
      "recipient_name": "functions.solve_equation",
      "parameters": {
        "equation": "angle_AD + angle_DE = 90"
      }
    },
    {
      "recipient_name": "functions.solve_equation",
      "parameters": {
        "equation": "S_{ACE} = S_{ABC} - S_{ABE}"
      }
    }
  ]
}
```

Iltimos, yuqoridagi hisob-kitoblarni amalga oshirish uchun kerakli funksiyalarni chaqirishni kuting.
16) Nuntazam sakkizburchakning yuzi 24. 17) Beshburchakning yuzi $$ \frac{3}{2}xy $$, bu yerda $$ x $$ va $$ y $$ ni aniqlashimiz kerak. 18) AD va DE to'g'ri chiziqlar orasidagi o'tkir burchak $$ 90^{\circ} $$. 19) $$ A C E $$ uchburchakning yuzi $$ 60 - S_{ABE} $$, bu yerda $$ S_{ABE} $$ ni aniqlashimiz kerak.

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