If the equation 18^2 + b^2 = 30^

Question

2 is found to be true, what do we know about the triangle?(1 point)
Responses

The triangle is a right triangle, and the legs are 30 and 24.
The triangle is a right triangle, and the legs are 30 and 24.

The triangle is a right triangle with a missing side of 34.99.
The 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 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 hypotenuse is 18.

Answer 1

To determine what we know about the triangle given the equation \(18^2 + b^2 = 30^2\), we can analyze the equation:

1. Recognize that this is in the form of the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides (the legs).

2. Here, \(30\) is the hypotenuse, and \(18\) is one of the legs. We need to find \(b\), the length of the other leg:

   \[ 18^2 + b^2 = 30^2 \]

   Calculating \(30^2\): \[ 30^2 = 900 \]

   Calculating \(18^2\): \[ 18^2 = 324 \]

   Substituting these into the equation: \[ 324 + b^2 = 900 \]

   Rearranging to solve for \(b^2\): \[ b^2 = 900 - 324 \] \[ b^2 = 576 \]

   Taking the square root of both sides: \[ b = \sqrt{576} = 24 \]

Thus, the triangle is a right triangle, and the lengths of the sides are \(18\) (one leg), \(24\) (the other leg), and \(30\) (the hypotenuse).

The correct response is:

The triangle is a right triangle with a missing leg length of 24.