If the square of the length of the longest side of a triangle is equal to the sum of the squares of the Choose... other two sides, then the triangle is a right triangle.

It looks like you're referring to the **converse of the Pythagorean theorem**. Let's complete and clarify your statement: **Complete Statement:** *If the square of the length of the longest side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle.* **Explanation:** - **Pythagorean Theorem:** In a right-angled triangle, the square of the hypotenuse (the longest side opposite the right angle) is equal to the sum of the squares of the other two sides. Mathematically, if \( c \) is the hypotenuse and \( a \) and \( b \) are the other two sides, then: \[ c^2 = a^2 + b^2 \] - **Converse of the Pythagorean Theorem:** This states that if in a triangle, the square of one side equals the sum of the squares of the other two sides, then that triangle must be a right-angled triangle. In other words, if: \[ c^2 = a^2 + b^2 \] then the triangle with sides \( a \), \( b \), and \( c \) is a right triangle with the right angle opposite side \( c \). **Implications:** - **Verification:** This converse is often used to verify whether a given triangle is right-angled when the lengths of its sides are known. - **Uniqueness:** Only in right-angled triangles does this specific relationship between the sides hold true. **Example:** Suppose you have a triangle with sides of lengths 3 units, 4 units, and 5 units. Check if it's a right triangle: \[ 5^2 = 25 \quad \text{and} \quad 3^2 + 4^2 = 9 + 16 = 25 \] Since \( 25 = 25 \), by the converse of the Pythagorean theorem, this is a right-angled triangle. **Conclusion:** Your statement is correct. The converse of the Pythagorean theorem provides a reliable method to determine whether a given triangle is right-angled based on the lengths of its sides.