Sylvia likes to walk to work when the weather is nice. She works at the bank at the corner of 41st Street and Edison Avenue. Her house is at the corner of 42nd Street and Johnson Avenue. She can walk down Johnson Avenue and turn right on 41st Street, or she can walk through the park. The distance from 42nd Street to 41st Street is 80 yards, and it is 60 yards from Johnson Avenue to Edison Avenue. Assuming the street intersections are 90° angles, how much shorter is it for Sylvia to walk through the park?
(1 point)
If Sylvia walks through the park instead of straight down Johnson Avenue and straight down 41st Street, the walk will be ____ yards shorter.
If Sylvia walks through the park, she will essentially be cutting diagonally across the block formed by 42nd Street, Johnson Avenue, 41st Street, and Edison Avenue. This forms a right triangle, where the distance along 42nd Street is the base, the distance along Johnson Avenue is the height, and the diagonal across the block is the hypotenuse.
Using the Pythagorean theorem, we can find the length of the hypotenuse.
The square of the hypotenuse is equal to the sum of the squares of the base and height:
Hypotenuse^2 = Base^2 + Height^2
Hypotenuse^2 = 80^2 + 60^2
Hypotenuse^2 = 6400 + 3600
Hypotenuse^2 = 10000
Hypotenuse = √10000
Hypotenuse = 100 yards
So, the distance Sylvia will walk through the park is 100 yards.
If Sylvia walks straight down Johnson Avenue and turns right on 41st Street, she will walk a distance of 80 + 60 = 140 yards.
Therefore, if Sylvia walks through the park instead of going straight down Johnson Avenue and 41st Street, the walk will be 140 - 100 = 40 yards shorter. Answer: \boxed{40}.