Westonci.ca is your trusted source for accurate answers to all your questions. Join our community and start learning today! Connect with a community of professionals ready to provide precise solutions to your questions quickly and accurately. Our platform provides a seamless experience for finding reliable answers from a network of experienced professionals.
Sagot :
To determine whether a triangle with sides [tex]\(\sqrt{95}\)[/tex] feet, 8 feet, and [tex]\(\sqrt{150}\)[/tex] feet is a right triangle, we need to test if these sides satisfy the Pythagorean theorem. The Pythagorean theorem states that, for a triangle to be a right triangle, the sum of the squares of the lengths of the two shorter sides (legs) must be equal to the square of the length of the longest side (hypotenuse).
We are given the sides:
- Side 1: [tex]\(\sqrt{95}\)[/tex] feet,
- Side 2: 8 feet,
- Side 3: [tex]\(\sqrt{150}\)[/tex] feet.
First, let's convert these into their approximate decimal values:
- [tex]\(\sqrt{95} \approx 9.7468\)[/tex] feet,
- Side 2 = 8 feet,
- [tex]\(\sqrt{150} \approx 12.2474\)[/tex] feet.
Next, we will proceed with the Pythagorean theorem in each possible configuration, verifying if the sum of the squares of any two sides equals the square of the third side.
### Case 1: Assume sides [tex]\(\sqrt{95}\)[/tex] and 8 are the legs, and [tex]\(\sqrt{150}\)[/tex] is the hypotenuse.
1. Calculate [tex]\((\sqrt{95})^2 + 8^2\)[/tex]:
[tex]\[ (\sqrt{95})^2 = 95 \][/tex]
[tex]\[ 8^2 = 64 \][/tex]
[tex]\[ 95 + 64 = 159 \][/tex]
2. Calculate [tex]\((\sqrt{150})^2\)[/tex]:
[tex]\[ (\sqrt{150})^2 = 150 \][/tex]
Since [tex]\(159 \neq 150\)[/tex], this does not satisfy the Pythagorean theorem.
### Case 2: Assume sides [tex]\(\sqrt{95}\)[/tex] and [tex]\(\sqrt{150}\)[/tex] are the legs, and 8 is the hypotenuse.
1. Calculate [tex]\((\sqrt{95})^2 + (\sqrt{150})^2\)[/tex]:
[tex]\[ (\sqrt{95})^2 = 95 \][/tex]
[tex]\[ (\sqrt{150})^2 = 150 \][/tex]
[tex]\[ 95 + 150 = 245 \][/tex]
2. Calculate [tex]\(8^2\)[/tex]:
[tex]\[ 8^2 = 64 \][/tex]
Since [tex]\(245 \neq 64\)[/tex], this does not satisfy the Pythagorean theorem either.
### Case 3: Assume sides 8 and [tex]\(\sqrt{150}\)[/tex] are the legs, and [tex]\(\sqrt{95}\)[/tex] is the hypotenuse.
1. Calculate [tex]\(8^2 + (\sqrt{150})^2\)[/tex]:
[tex]\[ 8^2 = 64 \][/tex]
[tex]\[ (\sqrt{150})^2 = 150 \][/tex]
[tex]\[ 64 + 150 = 214 \][/tex]
2. Calculate [tex]\((\sqrt{95})^2\)[/tex]:
[tex]\[ (\sqrt{95})^2 = 95 \][/tex]
Since [tex]\(214 \neq 95\)[/tex], this configuration does not satisfy the Pythagorean theorem either.
Since no configuration of the given sides satisfies the Pythagorean theorem, Gerard concluded that the triangle with sides [tex]\(\sqrt{95}\)[/tex] feet, 8 feet, and [tex]\(\sqrt{150}\)[/tex] feet is not a right triangle. Therefore, it cannot be used as a building frame support if a right triangle is necessary for the structure.
We are given the sides:
- Side 1: [tex]\(\sqrt{95}\)[/tex] feet,
- Side 2: 8 feet,
- Side 3: [tex]\(\sqrt{150}\)[/tex] feet.
First, let's convert these into their approximate decimal values:
- [tex]\(\sqrt{95} \approx 9.7468\)[/tex] feet,
- Side 2 = 8 feet,
- [tex]\(\sqrt{150} \approx 12.2474\)[/tex] feet.
Next, we will proceed with the Pythagorean theorem in each possible configuration, verifying if the sum of the squares of any two sides equals the square of the third side.
### Case 1: Assume sides [tex]\(\sqrt{95}\)[/tex] and 8 are the legs, and [tex]\(\sqrt{150}\)[/tex] is the hypotenuse.
1. Calculate [tex]\((\sqrt{95})^2 + 8^2\)[/tex]:
[tex]\[ (\sqrt{95})^2 = 95 \][/tex]
[tex]\[ 8^2 = 64 \][/tex]
[tex]\[ 95 + 64 = 159 \][/tex]
2. Calculate [tex]\((\sqrt{150})^2\)[/tex]:
[tex]\[ (\sqrt{150})^2 = 150 \][/tex]
Since [tex]\(159 \neq 150\)[/tex], this does not satisfy the Pythagorean theorem.
### Case 2: Assume sides [tex]\(\sqrt{95}\)[/tex] and [tex]\(\sqrt{150}\)[/tex] are the legs, and 8 is the hypotenuse.
1. Calculate [tex]\((\sqrt{95})^2 + (\sqrt{150})^2\)[/tex]:
[tex]\[ (\sqrt{95})^2 = 95 \][/tex]
[tex]\[ (\sqrt{150})^2 = 150 \][/tex]
[tex]\[ 95 + 150 = 245 \][/tex]
2. Calculate [tex]\(8^2\)[/tex]:
[tex]\[ 8^2 = 64 \][/tex]
Since [tex]\(245 \neq 64\)[/tex], this does not satisfy the Pythagorean theorem either.
### Case 3: Assume sides 8 and [tex]\(\sqrt{150}\)[/tex] are the legs, and [tex]\(\sqrt{95}\)[/tex] is the hypotenuse.
1. Calculate [tex]\(8^2 + (\sqrt{150})^2\)[/tex]:
[tex]\[ 8^2 = 64 \][/tex]
[tex]\[ (\sqrt{150})^2 = 150 \][/tex]
[tex]\[ 64 + 150 = 214 \][/tex]
2. Calculate [tex]\((\sqrt{95})^2\)[/tex]:
[tex]\[ (\sqrt{95})^2 = 95 \][/tex]
Since [tex]\(214 \neq 95\)[/tex], this configuration does not satisfy the Pythagorean theorem either.
Since no configuration of the given sides satisfies the Pythagorean theorem, Gerard concluded that the triangle with sides [tex]\(\sqrt{95}\)[/tex] feet, 8 feet, and [tex]\(\sqrt{150}\)[/tex] feet is not a right triangle. Therefore, it cannot be used as a building frame support if a right triangle is necessary for the structure.
We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. We hope our answers were useful. Return anytime for more information and answers to any other questions you have. Westonci.ca is your go-to source for reliable answers. Return soon for more expert insights.
