To determine whether the triangle with sides [tex]\(\sqrt{95}\)[/tex] feet, 8 feet, and [tex]\(\sqrt{150}\)[/tex] feet is a right triangle, we use the Pythagorean theorem. According to the Pythagorean theorem, a triangle is a right triangle if the square of the length of the longest side (the hypotenuse) is equal to the sum of the squares of the lengths of the other two sides.
Let's denote the sides of the triangle as follows:
- [tex]\( a = \sqrt{95} \)[/tex]
- [tex]\( b = 8 \)[/tex]
- [tex]\( c = \sqrt{150} \)[/tex]
To apply the Pythagorean theorem, we first need to identify the longest side. Let's find the values of the square roots:
- [tex]\( \sqrt{95} \approx 9.7468 \)[/tex]
- [tex]\( \sqrt{150} \approx 12.2474 \)[/tex]
So, we have the sides approximately as:
- Side 1: [tex]\( 8 \)[/tex]
- Side 2: [tex]\( 9.7468 \)[/tex]
- Side 3: [tex]\( 12.2474 \)[/tex]
The longest side is [tex]\( \sqrt{150} \approx 12.2474 \)[/tex], hence this will be considered as the hypotenuse [tex]\( c \)[/tex].
Next, we need to check if the Pythagorean theorem holds by verifying if:
[tex]\[ a^2 + b^2 = c^2 \][/tex]
Calculate the squares of the sides:
- [tex]\( a^2 = (\sqrt{95})^2 = 95 \)[/tex]
- [tex]\( b^2 = 8^2 = 64 \)[/tex]
- [tex]\( c^2 = (\sqrt{150})^2 = 150 \)[/tex]
Now, add the squares of the two shorter sides:
[tex]\[ a^2 + b^2 = 95 + 64 = 159 \][/tex]
Compare this to [tex]\( c^2 \)[/tex]:
[tex]\[ c^2 = 150 \][/tex]
Since [tex]\( 159 \)[/tex] is not equal to [tex]\( 150 \)[/tex], the Pythagorean theorem does not hold. Therefore, the triangle with sides [tex]\(\sqrt{95}\)[/tex] feet, 8 feet, and [tex]\(\sqrt{150}\)[/tex] feet is not a right triangle.
Thus, Gerard concluded correctly that the triangle cannot be used as a building frame support since it is not a right triangle. The triangle does not satisfy the condition for being a right triangle as per the Pythagorean theorem.
