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 properties of the triangle based on the equation \( 18^{2}+b^{2}=30^{2} \), we can analyze the equation step by step. 1. **Identify the components of the equation**: - The equation is in the form of the Pythagorean theorem, which states that for a right triangle with legs \( a \) and \( b \), and hypotenuse \( c \), the relationship is given by: \[ a^2 + b^2 = c^2 \] - In this case, we have \( a = 18 \), \( b \) is the unknown leg, and \( c = 30 \). 2. **Calculate \( 30^2 \)**: \[ 30^2 = 900 \] 3. **Calculate \( 18^2 \)**: \[ 18^2 = 324 \] 4. **Substitute and solve for \( b^2 \)**: \[ 324 + b^2 = 900 \] \[ b^2 = 900 - 324 \] \[ b^2 = 576 \] 5. **Find \( b \)**: \[ b = \sqrt{576} = 24 \] Now that we have determined that \( b = 24 \), we can conclude the following: - The triangle is a right triangle because it satisfies the Pythagorean theorem. - The lengths of the sides are \( 18 \) (one leg), \( 24 \) (the other leg), and \( 30 \) (the hypotenuse). Based on the options provided, the correct answer is: **The triangle is a right triangle with a missing leg length of 24.**