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.
