Math help with open ended... Tangent/secent lines and circles

Question

The courtyard shown in the figure below has a main walkway that runs north to south along the diameter of the courtyard. A crossing path is perpendicular to the main walkway and is 42 feet in length. The north end of the main walkway has a canopy cover. The length of the main walkway under the canopy is 20 feet.

Based on this information, what is the total length of the main walkway from the north end to the south end?

Answer 1

Based on your description, the radius "r" of the circle is cut into two parts by the perpendicular path. 20 feet lies north of the perpendicular path and the remaining portion (r-20) is the distance between the center and the perpendicular path.

Since the perpendicular path is 42 feet, 21 feet lies to the east and west of the main walkway.

With these two pieces of information, you can draw a right triangle with sides of r (as the hypotenuse), r-20, and 21. From here, you can use the Pythagorean Theorem to solve for "r" as follows:

r² = (r-20)² + 21²

r² = r² - 40r + 400 + 441

0 = -40r + 841

40r = 841

r = 841/40

The central walkway runs the diameter of the courtyard, so it must be twice as long as r. Therefore, the walkway is:

2 * 841/40 = 42.05 feet