If a right triangle has a leg that is 6 ft. long and the hypotenuse is 12 ft. long, how long is the other leg? Round to
the nearest tenth. (1 point)
Let's call the other leg x.
According to the Pythagorean theorem, in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs.
So, we can write the following equation:
x^2 + 6^2 = 12^2
Simplifying the equation, we have:
x^2 + 36 = 144
Subtracting 36 from both sides, we find:
x^2 = 108
Taking the square root of both sides, we have:
x ≈ √108
x ≈ 10.39
Rounding to the nearest tenth, the other leg is approximately 10.4 ft.
If a right triangle has a leg that is 20 ft. long and the hypotenuse is 30 ft. long, how long is the other leg? Round to the nearest hundredth.
Let's call the other leg x.
According to the Pythagorean theorem, in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs.
So, we can write the following equation:
x^2 + 20^2 = 30^2
Simplifying the equation, we have:
x^2 + 400 = 900
Subtracting 400 from both sides, we find:
x^2 = 500
Taking the square root of both sides, we have:
x ≈ √500
x ≈ 22.36
Rounding to the nearest hundredth, the other leg is approximately 22.36 ft.
Camila needs to fix the window on the front of her house. The bottom of the window is 16 ft. from the ground.
She only has a 20 ft. ladder. How far from the base of the house should she set up the ladder so that the top of
the ladder is exactly at the bottom of the window? (1 point)
Let's call the distance from the base of the ladder to the base of the house x.
Since the bottom of the window is 16 ft. from the ground, and she has a 20 ft. ladder, we can create a right triangle using the ladder as the hypotenuse.
According to the Pythagorean theorem, in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs.
So, we can write the following equation:
x^2 + 16^2 = 20^2
Simplifying the equation, we have:
x^2 + 256 = 400
Subtracting 256 from both sides, we find:
x^2 = 144
Taking the square root of both sides, we have:
x ≈ √144
x ≈ 12
Therefore, Camila should set up the ladder approximately 12 ft. from the base of the house to align the top of the ladder with the bottom of the window.
Sylvia likes to walk to work when the weather is nice. She works at the bank at the corner of 41 st 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 it is for Sylvia to walk through the park, we need to calculate the distance of each route.
The distance from 42nd Street to 41st Street is given as 80 yards.
The distance from Johnson Avenue to Edison Avenue is given as 60 yards.
If Sylvia walks straight down Johnson Avenue and then turns right on 41st Street, she will be walking along two straight sides of a right triangle. The distance she walks can be calculated using the Pythagorean theorem.
Let's call the distance she walks through the park x.
According to the Pythagorean theorem, in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs.
So, we can write the following equation:
x^2 + 80^2 = 60^2
Simplifying the equation, we have:
x^2 + 6400 = 3600
Subtracting 3600 from both sides, we find:
x^2 = 3600 - 6400
x^2 = 2800
Taking the square root of both sides, we have:
x ≈ √2800
x ≈ 52.92
Therefore, if Sylvia walks through the park instead of straight down Johnson Avenue and straight down 41st Street, the walk will be approximately 52.92 yards shorter.
wrong
My apologies for the mistake. Let's correct the solution.
We can consider the distance from Sylvia's house to the corner of 41st Street and Johnson Avenue as the two legs of a right triangle. The direct route along Johnson Avenue is one leg and the route through the park is the other leg.
The distance from 42nd Street to 41st Street is given as 80 yards.
The distance from Johnson Avenue to Edison Avenue is given as 60 yards.
To find the distance of the route through the park, we can use the Pythagorean theorem.
Let's call the distance of the route through the park x.
According to the Pythagorean theorem, in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs.
So, we can write the following equation:
x^2 = 80^2 + 60^2
Simplifying the equation, we have:
x^2 = 6400 + 3600
x^2 = 10000
Taking the square root of both sides, we have:
x ≈ √10000
x ≈ 100
Therefore, if Sylvia walks through the park instead of straight down Johnson Avenue and straight down 41st Street, the walk will be approximately 100 yards shorter.