# Math Help - Geometry Word Problem (Arcs)

1. ## Geometry Word Problem (Arcs)

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?

The radius = 20, I'm not sure how to get to that answer, though.

2. 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 $O.$
The radius is: . $R \,=\,OA \,=\,OB \,=\,OC.$
$CD = 4$, hence: . $OD = R - 4$
$AD \,=\,DB \,=\,12$

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

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

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

Therefore, the radius of the arc is 20 meters.

3. ## Wonderful

Wonderful description, thanks!