Question
Allie wants to make sure that the pieces of cloth for a costume are right triangles.
A right triangle with the base labeled 12 inches, the hypotenuse labeled 13 inches, and the opposite side labeled 5 inches.
Determine whether the triangle is a right triangle.
A right triangle with the base labeled 12 inches, the hypotenuse labeled 13 inches, and the opposite side labeled 5 inches.
Determine whether the triangle is a right triangle.
Answers
GPT-4o mini
To determine whether the triangle is a right triangle, we can use the Pythagorean theorem. The theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.
In this case, we have:
- Base (one leg) \( a = 12 \) inches
- Opposite side (the other leg) \( b = 5 \) inches
- Hypotenuse \( c = 13 \) inches
We need to check whether the following holds true:
\[
c^2 = a^2 + b^2
\]
Calculating each side, we have:
- \( c^2 = 13^2 = 169 \)
- \( a^2 = 12^2 = 144 \)
- \( b^2 = 5^2 = 25 \)
Now sum \( a^2 \) and \( b^2 \):
\[
a^2 + b^2 = 144 + 25 = 169
\]
Now, since \( c^2 = a^2 + b^2 \):
\[
169 = 169
\]
This equality confirms that the triangle satisfies the Pythagorean theorem. Therefore, the triangle is indeed a right triangle.
In this case, we have:
- Base (one leg) \( a = 12 \) inches
- Opposite side (the other leg) \( b = 5 \) inches
- Hypotenuse \( c = 13 \) inches
We need to check whether the following holds true:
\[
c^2 = a^2 + b^2
\]
Calculating each side, we have:
- \( c^2 = 13^2 = 169 \)
- \( a^2 = 12^2 = 144 \)
- \( b^2 = 5^2 = 25 \)
Now sum \( a^2 \) and \( b^2 \):
\[
a^2 + b^2 = 144 + 25 = 169
\]
Now, since \( c^2 = a^2 + b^2 \):
\[
169 = 169
\]
This equality confirms that the triangle satisfies the Pythagorean theorem. Therefore, the triangle is indeed a right triangle.