Question
Gerard concluded that the triangle with sides StartRoot 95 EndRoot feet, 8 feet, and StartRoot 150 EndRoot cannot be used as a building frame support on the house because it is not a right triangle.
Answers
GPT-4o mini
To determine whether the triangle with sides \( \sqrt{95} \) feet, 8 feet, and \( \sqrt{150} \) feet is a right triangle, we can use the Pythagorean theorem. According to the theorem, for a triangle to be a right triangle, the square of the length of the longest side must equal the sum of the squares of the lengths of the other two sides.
1. First, identify the lengths of the sides:
- \( a = \sqrt{95} \)
- \( b = 8 \)
- \( c = \sqrt{150} \)
2. Determine the longest side. Since \( \sqrt{150} \) is greater than \( \sqrt{95} \) and 8, we will consider \( c = \sqrt{150} \) as the longest side.
3. According to the Pythagorean theorem, we will check if:
\[
c^2 = a^2 + b^2
\]
4. Calculate each square:
- \( c^2 = \sqrt{150}^2 = 150 \)
- \( a^2 = \sqrt{95}^2 = 95 \)
- \( b^2 = 8^2 = 64 \)
5. Now, substitute the squares into the equation:
\[
150 \stackrel{?}{=} 95 + 64
\]
\[
150 \stackrel{?}{=} 159
\]
6. Since 150 is not equal to 159, the Pythagorean theorem does not hold true.
Hence, the triangle with sides \( \sqrt{95} \), 8, and \( \sqrt{150} \) is not a right triangle. Gerard's conclusion about the triangle's suitability as a building frame support is correct; it cannot be used for that purpose if a right triangle is required.
1. First, identify the lengths of the sides:
- \( a = \sqrt{95} \)
- \( b = 8 \)
- \( c = \sqrt{150} \)
2. Determine the longest side. Since \( \sqrt{150} \) is greater than \( \sqrt{95} \) and 8, we will consider \( c = \sqrt{150} \) as the longest side.
3. According to the Pythagorean theorem, we will check if:
\[
c^2 = a^2 + b^2
\]
4. Calculate each square:
- \( c^2 = \sqrt{150}^2 = 150 \)
- \( a^2 = \sqrt{95}^2 = 95 \)
- \( b^2 = 8^2 = 64 \)
5. Now, substitute the squares into the equation:
\[
150 \stackrel{?}{=} 95 + 64
\]
\[
150 \stackrel{?}{=} 159
\]
6. Since 150 is not equal to 159, the Pythagorean theorem does not hold true.
Hence, the triangle with sides \( \sqrt{95} \), 8, and \( \sqrt{150} \) is not a right triangle. Gerard's conclusion about the triangle's suitability as a building frame support is correct; it cannot be used for that purpose if a right triangle is required.