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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°
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 how much shorter Sylvia's walk is when she goes through the park instead of walking straight down Johnson Avenue and then 41st Street, we need to calculate both routes.

Route A (Walking straight down the streets):

1. From her house at the corner of 42nd Street and Johnson Avenue, she walks down Johnson Avenue to the bank at the corner of 41st Street and Edison Avenue:
   • Distance along Johnson Avenue = 80 yards
2. She then turns right and walks down 41st Street to the bank:
   • Distance along 41st Street = 60 yards

Total Distance for Route A: \[ 80 \text{ yards} + 60 \text{ yards} = 140 \text{ yards} \]

Route B (Walking through the park):

• This route is the hypotenuse of the right triangle formed by the two sides:
  • One side is the distance from 42nd Street to 41st Street (80 yards).
  • The other side is the distance from Johnson Avenue to Edison Avenue (60 yards).

To calculate the hypotenuse (the park), we use the Pythagorean theorem: \[ c = \sqrt{a^2 + b^2} \] where \( a = 80 \) yards and \( b = 60 \) yards.

Calculating the hypotenuse: \[ c = \sqrt{(80)^2 + (60)^2} \] \[ c = \sqrt{6400 + 3600} \] \[ c = \sqrt{10000} \] \[ c = 100 \text{ yards} \]

Total Distance for Route B:

• The distance through the park is 100 yards.

Difference in Distance: To find out how much shorter Route B (through the park) is compared to Route A (down the streets): \[ \text{Difference} = \text{Distance A} - \text{Distance B} \] \[ \text{Difference} = 140 \text{ yards} - 100 \text{ yards} = 40 \text{ yards} \]

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