Hello, Gui!

Two parallel chords are 16 cm and 30 cm long are 23 cm apart.

Find the radius of the cricle. Code:

* * *
* 8 *
C * - - - + - - - * D
* | * *
23-x | * r
* | * *
* O* *
* | * r *
x| *
A *- - - - + - - - -* B
* 15 *
* *
* * *

The longer chord is: .$\displaystyle AB = 30$

. . It is $\displaystyle x$ cm from the center $\displaystyle O.$

The shorter chord is: .$\displaystyle CD = 16$

. . It is $\displaystyle 23-x$ cm from the center.

Let $\displaystyle r$ = the radius: .$\displaystyle r = OA = OB = OC = OD.$

In the two right triangles we have:

. . $\displaystyle \begin{array}{ccccc}x^2 + 15^2 &=& r^2 & [1] \\

(23-x)^2 + 8^2 &=& r^2 & [2] \end{array}$

Equate [1] and [2]: .$\displaystyle x^2 + 15^2 \:=\:(23-x)^2 + 8^2 \quad\Rightarrow\quad x = 8$

Substitute into [1]: .$\displaystyle 8^2 + 15^2 \:=\:r^2 \quad\Rightarrow\quad \boxed{r \:=\:17} $