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.
To find out how much shorter the walk would be through the park, we need to find the length of the hypotenuse joining the bank and the vertex of Johnson Avenue and 42nd street.
Since the intersection of 42nd Street and Johnson Avenue forms a right angle, it can be considered as the vertex of a right triangle with legs 60 yards and 80 yards.
Using the Pythagorean theorem, we can calculate the length of the hypotenuse (park):
park = √(60^2 + 80^2)
= √(3600 + 6400)
= √10000
= 100 yards
So, if Sylvia walks through the park instead of straight down Johnson Avenue and straight down 41st Street, the walk will be 100 yards shorter.
To find out how much shorter it is for Sylvia to walk through the park, we need to calculate the length of the diagonal from the corner of 42nd Street and Johnson Avenue to the corner of 41st Street and Edison Avenue.
We can use the Pythagorean theorem to find the length of this diagonal.
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 diagonal of the rectangle is the hypotenuse and the lengths of the sides are 80 yards and 60 yards.
Using the Pythagorean theorem, we can calculate the length of the diagonal:
d² = 80² + 60²
d² = 6400 + 3600
d² = 10000
d = √10000
d = 100 yards
Therefore, the distance from the corner of 42nd Street and Johnson Avenue to the corner of 41st Street and Edison Avenue is 100 yards.
If Sylvia walks through the park instead of straight down Johnson Avenue and straight down 41st Street, the walk will be 100 yards shorter.
To determine how much shorter the walk is through the park, we first need to find the length of the hypotenuse joining the vertex of Johnson Avenue and 42nd street to the bank at the vertex of 41st street and Edison Avenue. This length is labeled as "park" in the image.
Based on 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.
Since the rectangle is formed from two right triangles, we can use the Pythagorean theorem to find the length of the hypotenuse:
c^2 = a^2 + b^2
where c represents the hypotenuse, and a and b represent the two sides of the right triangle.
In this case, the hypotenuse (c) is the length of the "park" line, and the two sides (a and b) are the lengths of Johnson Avenue (80 yards) and 41st street (60 yards), respectively.
Using the Pythagorean theorem, we can calculate:
park^2 = Johnson Avenue^2 + 41st street^2
park^2 = 80^2 + 60^2
park^2 = 6400 + 3600
park^2 = 10000
To find the length of the hypotenuse (park), we take the square root of both sides:
park = √10000
park = 100 yards
Therefore, if Sylvia walks through the park instead of straight down Johnson Avenue and straight down 41st Street, the walk will be 100 yards shorter.