What is the radius of this circle?

A bridge over a river has the shape of a circular arc. The span of the bridge is 24 meters. The midpoint of the arc is 4 meters higher than the endpoints.

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The flat part of the bridge is a chord of the circle of which the arc is a part. The chord length is 24m, and the distance from the midpoint of the chord to the circle is 4m. We are asked to find the radius of the circle.

From geometry we know that if two chords of a circle intersect, then the segments cut by the intersection are proportional. In other words, if the first chord is cut into segments of length r and t, and the second chord is cut into segments of length u and v, then rt=uv.

We also know that a radius drawn perpendicular to a chord bisects the chord and its corresponding arc.

Draw a circle with center O, and chord AB. The midpoint of AB is M, and the midpoint of arc AB is N. Then ON is the radius we seek.

From the chord-chord length theorem we have AM*MB=NM*(2MO+MN). Substituting the known values yields 12*12=4(2MO+4).

Solving for MO yields MO=16. Then the radius is MO+MN=16+4=20.

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**The radius of the circle is 20m.**

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