Hello, dvh9!

A bridge over a river has a shape of a circular arc.

The span of the bridge is 24 meters. (The span is the length of the chord of the arc.)

The midpoint of the arc is 4 meters higher than the endpoints.

What is the radius of the circle that contains this arc? Code:

C
* * *
* 4| *
A *-------o-------* B
* * 12 |D * *
* | * R
* * | * *
* - - - - * - - - - *
O

The center of the circle is $\displaystyle O.$

The radius is: .$\displaystyle R \,=\,OA \,=\,OB \,=\,OC.$

$\displaystyle CD = 4$, hence: .$\displaystyle OD = R - 4$

$\displaystyle AD \,=\,DB \,=\,12$

In right triangle $\displaystyle ODB\!:\;\;OD^2 + DB^2 \:=\:OB^2$

So we have: .$\displaystyle (R-4)^2 + 12^2 \:=\:R^2 \quad\Rightarrow\quad R^2 - 8R + 16 + 144 \:=\:R^2$

. . $\displaystyle -8R \:=\:-160 \quad\Rightarrow\quad R \:=\:20$

Therefore, the radius of the arc is 20 meters.