Question

A bridge over a river has the 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

Answer 1

20 m

Explanation:

radius 'r', half the chord, and dist from center to the chord has formed a right angle triangle

where,

r is hypotenuse

(r-4) = one side i.e dist from center to chord)

12 = other side i.e half the length of the chord)

By applying Pythagoras theorem, we will have

r² = 12² + (r-4)²

r²= 144 + r² - 8r + 16 --->(cancel out r²)

0 = -8r + 160

8r = 160

r = 160/8

r = 20 m

Thus, the radius of the circle containing this arc is 20m

You can also verify this :

distance from center to chord: 20 - 4 = 16

r =
`\sqrt{16^(2)+12^(2) }`

r = 20

Answer 2

20 meters

Explanation:

Seeing the image attached, we can observe that there is a triangle formed by half of the chord of the arc, the radius (hypotenusa) and part of the radius (r-4). Using pythagoras theorem in this triangle, we have:

(r-4)^2 + 12^2 = r^2

r^2 - 8r + 16 + 144 = r^2

8r = 160

r = 20

So the radius of the circle that contains this arc is 20 meters.