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?
If Sylvia walks through the park instead of straight down Johnson Avenue and straight down 41st Street, the walk will be ___ yards shorter.
To find how much shorter it is for Sylvia to walk through the park, we need to calculate the length of the hypotenuse joining the bank and the vertex of Johnson Avenue and 42nd street.
Since we have a right triangle formed by the hypotenuse (park), Johnson Avenue (base), and 42nd Street (height), we can use the Pythagorean theorem.
The length of Johnson Avenue is 80 yards, and the length of 42nd Street is 60 yards.
Let's label the length of the hypotenuse as "c".
According to the Pythagorean theorem, we have the equation: c^2 = 80^2 + 60^2
Simplifying the equation, we get: c^2 = 6400 + 3600
c^2 = 10000
Taking the square root of both sides, we get: c = √10000
c = 100 yards
Therefore, the length of the hypotenuse (park) is 100 yards.
To find how much shorter it is to walk through the park, we need to calculate the difference between walking down Johnson Avenue and 41st Street (80 + 60 = 140 yards) and walking through the park (100 yards).
140 - 100 = 40
Therefore, it is 40 yards shorter for Sylvia to walk through the park. Answer: \boxed{40}.
wrong
I apologize for the incorrect answer. Let's recalculate.
The length of Johnson Avenue is 80 yards, and the length of 42nd Street is 60 yards. Since we are looking for the length of the hypotenuse (park), we can use the Pythagorean theorem.
Let's label the length of the hypotenuse as "c". According to the Pythagorean theorem, we have the equation:
c^2 = 80^2 + 60^2
c^2 = 6400 + 3600
c^2 = 10000
Taking the square root of both sides, we get:
c = √10000
c = 100 yards
Therefore, the length of the hypotenuse (park) is 100 yards.
To find out how much shorter it is for Sylvia to walk through the park, we need to calculate the difference between walking down Johnson Avenue and 41st Street (80 + 60 = 140 yards) and walking through the park (100 yards):
140 - 100 = 40
Therefore, it is 40 yards shorter for Sylvia to walk through the park. Answer: \boxed{40}.
To find out how much shorter it is for Sylvia to walk through the park, we need to calculate the length of the hypotenuse joining the bank and the intersection of Johnson Avenue and 42nd Street.
We can use the Pythagorean theorem to find the length of the hypotenuse. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.
In this case, one side of the right triangle is the distance from the bank to the intersection of 41st Street and Edison Avenue, which is 60 yards. The other side is the distance from the intersection of Johnson Avenue and 42nd Street to the bank, which is 80 yards.
Let's call the hypotenuse "park" and use the Pythagorean theorem to calculate its length:
park^2 = (60^2) + (80^2)
park^2 = 3600 + 6400
park^2 = 10000
To find the length of "park," we take the square root of both sides:
park = √10000
park = 100 yards
So the hypotenuse "park" measures 100 yards.
Now, we can calculate the shorter distance by subtracting the length of Johnson Avenue (80 yards) from the length of the hypotenuse (100 yards):
Shorter distance = park - Johnson Avenue
Shorter distance = 100 - 80
Shorter distance = 20 yards
Therefore, if Sylvia walks through the park instead of straight down Johnson Avenue and straight down 41st Street, the walk will be 20 yards shorter.