Use the image to answer the question.

Question

A rectangle is formed from two right triangles. The base of the rectangle measures 80 yards and is labeled Johnson Avenue. The top of the rectangle is labeled Edison Avenue. The left side joining both the avenues is labeled 41st street and measures 60 yards. The right side of the rectangle is labeled 42nd street. A bank is located at the vertex of 41st street and Edison Avenue. The hypotenuse joining the bank and the vertex of Johnson Avenue and 42nd street is labeled park.

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.

Answer 1

To find out how much shorter it is for Sylvia to walk through the park instead of walking down Johnson Avenue and then turning right onto 41st Street, we need to calculate both distances.

1. Distance walking straight down Johnson Avenue and then turning on 41st Street:

   • Johnson Avenue is 80 yards.
   • 41st Street is 60 yards.

   Total distance = 80 yards + 60 yards = 140 yards.

2. Distance walking through the park (the hypotenuse):

   • We can visualize this scenario as a right triangle where:
     • The base (horizontal distance) is 80 yards (from 42nd Street to 41st Street).
     • The height (vertical distance) is 60 yards (from Johnson Avenue to Edison Avenue).
   • We can apply the Pythagorean theorem to calculate the hypotenuse:

   \[ c^2 = a^2 + b^2 \] \[ c^2 = 80^2 + 60^2 \] \[ c^2 = 6400 + 3600 \] \[ c^2 = 10000 \] \[ c = \sqrt{10000} = 100 \text{ yards} \]

3. Finding the difference in distance:

   • Distance walking through the park = 100 yards.
   • Distance walking straight down = 140 yards.

   Difference = 140 yards - 100 yards = 40 yards.

Thus, if Sylvia walks through the park instead of straight down Johnson Avenue and straight down 41st Street, the walk will be 40 yards shorter.