19) $$ A B C $$ uchburchak yuzasi 60 ga teng. $$ \begin{array}{l}\overrightarrow{B E}=-\frac{1}{2} \overrightarrow{B A} ext { bo'lsa, } \ A C E ext { uchburchak yuzini toping. }\end{array} $$

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19) $$ A B C $$ uchburchak yuzasi 60 ga teng. $$ \begin{array}{l}\overrightarrow{B E}=-\frac{1}{2} \overrightarrow{B A} 	ext { bo'lsa, } \ A C E 	ext { uchburchak yuzini toping. }\end{array} $$

UpStudy AI · Accepted Answer

Uchburchak ABC ning yuzasi 60 ga teng. Bizga berilgan:
$$
\overrightarrow{BE} = -\frac{1}{2} \overrightarrow{BA}
$$
Bu yerda E nuqtasi BA bo‘ylab B nuqtasi tomonida joylashgan va BA vektorining yarmi bilan teskari yo‘nalishda joylashgan.

**Qadamlar:**

1. **Koordinatalarni Belgilash:**
   - B nuqtasini koordinatalar tizimining boshlang‘ich nuqtasi sifatida qabul qilamiz: $$ B(0, 0) $$.
   - $$ \overrightarrow{BA} = \mathbf{a} $$ vektorini x o‘qi bo‘yicha yo‘lni olgan deb tasavvur qilamiz: $$ A(a, 0) $$.
   - $$ \overrightarrow{BC} = \mathbf{c} $$ vektorini y o‘qi bo‘yicha yo‘lni olgan deb qabul qilamiz: $$ C(0, b) $$.

2. **Nuqta E ning Koordinatalarini Topish:**
   $$
   \overrightarrow{BE} = -\frac{1}{2} \overrightarrow{BA} = -\frac{1}{2} \mathbf{a} \implies E\left(-\frac{a}{2}, 0\right)
   $$

3. **ABC Uchburchagini Qayta Ko‘rib Chiqish:**
   - ABC uchburchagining yuzi:
     $$
     S_{ABC} = \frac{1}{2} \times |a \times b| = 60 \implies a \times b = 120
     $$

4. **ACE Uchburchagi Yuzasini Hisoblash:**
   - A, C, E nuqtalarining koordinatalari:
     $$
     A(a, 0), \quad C(0, b), \quad E\left(-\frac{a}{2}, 0\right)
     $$
   - ACE uchburchagining yuzi:
     $$
     S_{ACE} = \frac{1}{2} \times | \vec{AC} \times \vec{AE} |
     $$
     $$
     \vec{AC} = \left( -a, b \right), \quad \vec{AE} = \left( -\frac{3a}{2}, 0 \right)
     $$
     $$
     \vec{AC} \times \vec{AE} = (-a) \times 0 - b \times \left(-\frac{3a}{2}\right) = \frac{3ab}{2}
     $$
     $$
     S_{ACE} = \frac{1}{2} \times \frac{3ab}{2} = \frac{3ab}{4}
     $$
     $$
     \text{Bizda } ab = 120 \text{ bo‘lgani uchun: }
     $$
     $$
     S_{ACE} = \frac{3 \times 120}{4} = 90
     $$

**Javob:**
ACE uchburchagining yuzi **90 ga teng**.
ACE uchburchagining yuzi 90 ga teng.

uchburchak yuzasi 60 ga teng.

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