Aylananing AB=12 vatari uchlaridan aylanaga o'tkazilgan urinmalar C nuqtada kesishadi. ABC uchburchak yuzi 54 ga teng. Aylana radiusini toping.

Berilgan: \(AB=12\), \(S_{ABC}=54\).

Topish kerak: aylana radiusi \(R\).

CA va CB — aylananing urinmalari. Shuning uchun \(CA=CB\), demak, \(ABC\) teng yonli uchburchak. Uchburchak balandligi AB vatarning o'rtasiga tushadi.

Uchburchak yuzidan balandlikni topamiz:

\(S=\frac{1}{2}\cdot AB\cdot h\).

\(54=\frac{1}{2}\cdot12\cdot h\), bundan \(h=9\).

Urinma va radius o'zaro perpendikulyar. Markazdan AB vatargacha bo'lgan masofa \(R-9\) ga teng. Vatar yarmining uzunligi \(6\), shuning uchun:

\(R^2=(R-9)^2+6^2\).

\(R^2=R^2-18R+81+36\).

\(18R=117\).

\(R=\frac{117}{18}=\frac{13}{2}=6.5\).

Javob: \(6.5\).