Pythagorean Theorem Proofs Quick Check If the equation \( 18^{2}+b^{2}=30^{2} \) is found to be true, what do we know about the triangle? (1 point) OThe triangle is a right triangle with a missing side of 34.99 . The triangle is a right triangle with a missing leg length of 24 . The triangle is a right triangle, and the hypotenuse is 18 . The triangle is a right triangle, and the legs are 30 and 24 .

To determine the nature of the triangle using the Pythagorean Theorem, we start with the equation \( 18^{2} + b^{2} = 30^{2} \). Solving for \( b \) reveals that \( b^2 = 30^2 - 18^2 = 900 - 324 = 576 \), so \( b = 24 \). This confirms that the triangle is indeed a right triangle, with one leg measuring 18 and the other leg measuring 24, while the hypotenuse is 30. Remember, the Pythagorean Theorem applies exclusively to right triangles. This means that if you’re given two side lengths, always check to see if the third side you calculate makes sense based on this theorem. When working through problems, be cautious of miscalculating sides or misidentifying the hypotenuse—often the largest side!