# isosceles triangle inscribed in a circle, find the circle's radius....

• Feb 6th 2010, 06:54 AM
snigdha
isosceles triangle inscribed in a circle, find the circle's radius....
PQR is an isosceles http://searchdatacenter.techtarget.c...ages/delta.jpg ,inscribed in a circle with centre O, such that PQ=PR=13cm and QR=10cm. Find the radius of the circle.
• Feb 6th 2010, 07:22 AM
Wilmer
• Feb 6th 2010, 08:31 AM
Soroban
Hello, snigdha!

Quote:

Isosceles $\displaystyle \Delta PQR$ is inscribed in a circle with center $\displaystyle O$
such that: .$\displaystyle PQ=PR=13\text{ cm and }QR=10\text{ cm.}$
Find the radius of the circle.

Code:

                P               * o *           *    /|\    *         *    / | \    *       *    /  |r \    *             /  |  \ 13       *    /    |    \    *       *  /    O*  r \  *       *  /      |  *  \  *         /      |    * \     Q o- - - - o - - - -o R         *      S  5  *           *          *               * * *

In right triangle $\displaystyle PSR\!:\;PS^2 + 5^2 \:=\:13^2 \quad\Rightarrow\quad PS = 12$

. . Then: .$\displaystyle OS \,=\,12-r$

In right triangle $\displaystyle OSR\!:\;\;OS^2 + SR^2 \,=\,OR^2 \quad\Rightarrow\quad (12-r)^2 + 5^2 \:=\:r^2$

And we have: .$\displaystyle 144 - 24t + r^2 + 25 \:=\:r^2 \quad\Rightarrow\quad -24r \:=\:-169$

. . Therefore: .$\displaystyle r \;=\;\frac{169}{24}\text{ cm}$