Question
Given a triangle with side lengths of a, b, and c, where c is the longest side, which statement is true?
If b2 + c2 = a2, then the triangle is a right triangle.
If a2 + c2 = b2, then the triangle is a right triangle.
If a2 + b2 =c2, then the triangle is a right triangle.
If a2 = b2 = c2, then the triangle is a right triangle.
If b2 + c2 = a2, then the triangle is a right triangle.
If a2 + c2 = b2, then the triangle is a right triangle.
If a2 + b2 =c2, then the triangle is a right triangle.
If a2 = b2 = c2, then the triangle is a right triangle.
Answers
GPT-4o mini
In the context of a triangle with sides of lengths \( a \), \( b \), and \( c \) (where \( c \) is the longest side), the true statement in relation to the conditions for identifying a right triangle is:
**If \( a^2 + b^2 = c^2 \), then the triangle is a right triangle.**
This is known as the Pythagorean theorem, which states that for a right triangle, the square of the length of the hypotenuse (the longest side, \( c \)) is equal to the sum of the squares of the lengths of the other two sides (\( a \) and \( b \)).
The other statements are not correct:
- If \( b^2 + c^2 = a^2 \): This would imply that \( a \) is a hypotenuse and \( c \) is not the longest side, which contradicts the assumption that \( c \) is the longest side.
- If \( a^2 + c^2 = b^2 \): Similarly, this places \( b \) as the hypotenuse and contradicts the setup.
- If \( a^2 = b^2 = c^2 \): This condition means that all sides are equal in length, which describes an equilateral triangle, not necessarily a right triangle.
Therefore, the only valid statement regarding right triangles with the given conditions is:
**If \( a^2 + b^2 = c^2 \), then the triangle is a right triangle.**
**If \( a^2 + b^2 = c^2 \), then the triangle is a right triangle.**
This is known as the Pythagorean theorem, which states that for a right triangle, the square of the length of the hypotenuse (the longest side, \( c \)) is equal to the sum of the squares of the lengths of the other two sides (\( a \) and \( b \)).
The other statements are not correct:
- If \( b^2 + c^2 = a^2 \): This would imply that \( a \) is a hypotenuse and \( c \) is not the longest side, which contradicts the assumption that \( c \) is the longest side.
- If \( a^2 + c^2 = b^2 \): Similarly, this places \( b \) as the hypotenuse and contradicts the setup.
- If \( a^2 = b^2 = c^2 \): This condition means that all sides are equal in length, which describes an equilateral triangle, not necessarily a right triangle.
Therefore, the only valid statement regarding right triangles with the given conditions is:
**If \( a^2 + b^2 = c^2 \), then the triangle is a right triangle.**