Bridge question about radius

Question

A footbridge is in the shape of an arc of a circle. The bridge is 8 ft tall and 22 ft long, horizontally. What is the radius of the circle that contains the bridge? Round your answer to the nearest tenth.

Answer 1

Draw a picture!

First, draw a triangle-- one side on top, and the bottom angle pointing straight down. The top represents the length of the bridge---label it 22'. The 2 other (equal) sides are the radius of the circle of which the bridge arc is part. Sketch in the arc of the bridge, showing the top 8' above the top of the triangle. Get the sketch adjusted / redrawn such that the 8' bridge height is displayed as roughly 1/3 of the length of the bridge.

Finally, draw a vertical line from the bottom point to the bridge arc.

Once you have the right picture, you will have two similar right triangles. Each one will have the following dimensions:

Hypotenuse: R

Top Side: 11 (1/2 the length of the bridge)

Vertical side: R - 8

(R is the radius of the arc)

Next, use the Pythagorean Theorem to solve for R:

11² + ( R - 8 )² = R²

121 + R² - 16*R + 64 = R²

-16*R = -185

R = 11.6