Use the image to answer the 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.
Skip to navigation
page 11 of 12
To find out how much shorter it is for Sylvia to walk through the park, we need to determine the length of the hypotenuse labeled "park" in the image.
Using the Pythagorean theorem, we can 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, the lengths of the other two sides are 80 yards (Johnson Avenue) and 60 yards (41st Street).
So, using the Pythagorean theorem, we have:
Hypotenuse^2 = 80^2 + 60^2
Hypotenuse^2 = 6400 + 3600
Hypotenuse^2 = 10000
To solve for the Hypotenuse, we take the square root of both sides:
Hypotenuse = sqrt(10000)
Hypotenuse = 100 yards
Therefore, the length of the hypotenuse (park) is 100 yards.
Since Sylvia's house is at the corner of 42nd Street and Johnson Avenue, and she works at the corner of 41st Street and Edison Avenue, walking straight down Johnson Avenue and straight down 41st Street is like walking the sides of the rectangle.
The length of the side labeled "42nd street" is 80 yards, and the length of the side labeled "41st street" is 60 yards.
So, the total distance Sylvia would walk if she goes straight down Johnson Avenue and straight down 41st Street is 80 yards + 60 yards = 140 yards.
If Sylvia walks through the park, her walk would only be the length of the hypotenuse (park), which is 100 yards.
Therefore, if Sylvia walks through the park instead of straight down Johnson Avenue and straight down 41st Street, the walk will be 140 yards - 100 yards = 40 yards shorter.
To determine how much shorter it is for Sylvia to walk through the park, we need to find the length of the hypotenuse labeled "park" and compare it to the sum of the lengths of Johnson Avenue and 41st Street.
From the given information, we know that the base of the rectangle (Johnson Avenue) measures 80 yards and the left side (41st Street) measures 60 yards. We can use the Pythagorean theorem to find the length of the hypotenuse (park).
The Pythagorean theorem states that the square of the hypotenuse (c) is equal to the sum of the squares of the other two sides (a and b) in a right triangle. In this case, we can consider Johnson Avenue as side a and 41st Street as side b.
Applying the Pythagorean theorem:
c^2 = a^2 + b^2
park^2 = 80^2 + 60^2
park^2 = 6400 + 3600
park^2 = 10000
Taking the square root of both sides, we find:
park = √10000
park = 100 yards
So, the length of the hypotenuse "park" is 100 yards.
Now, we can compare this distance to the sum of the lengths of Johnson Avenue (80 yards) and 41st Street (60 yards):
80 + 60 = 140 yards
Therefore, if Sylvia walks through the park instead of straight down Johnson Avenue and 41st Street, the walk will be 140 - 100 = 40 yards shorter.
To find out how much shorter it is for Sylvia to walk through the park, we need to find the length of the hypotenuse joining the bank and the corner of 42nd Street and Johnson Avenue.
Using the Pythagorean theorem, we can calculate the length of the hypotenuse:
c^2 = a^2 + b^2
c^2 = 80^2 + 60^2
c^2 = 6400 + 3600
c^2 = 10000
c = √10000
c = 100
So the length of the hypotenuse (or the park) is 100 yards.
The distance from 42nd Street to 41st Street is 80 yards, so if Sylvia walks through the park instead of straight down Johnson Avenue and straight down 41st Street, her walk will be 80 - 100 = -20 yards shorter.