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

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

The triangle is a right triangle, and the hypotenuse is 18.
The triangle is a right triangle, and the hypotenuse is 18.

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

1 answer

The equation \( 18^2 + b^2 = 30^2 \) can be analyzed using the Pythagorean theorem, which states that in a right triangle, the sum of the squares of the lengths of the two legs (the sides adjacent to the right angle) is equal to the square of the length of the hypotenuse (the side opposite the right angle).

In the given equation:

  • \( 18^2 \) represents one leg of the triangle,
  • \( b^2 \) represents the other leg,
  • \( 30^2 \) represents the hypotenuse.

To find the value of \( b \):

  1. Calculate \( 30^2 \): \[ 30^2 = 900 \]

  2. Calculate \( 18^2 \): \[ 18^2 = 324 \]

  3. Substitute these values into the equation: \[ 324 + b^2 = 900 \]

  4. Solve for \( b^2 \): \[ b^2 = 900 - 324 \] \[ b^2 = 576 \]

  5. Taking the square root gives: \[ b = 24 \]

Thus, \( b \) is 24.

With this information, we can conclude that:

  • The triangle is a right triangle,
  • The lengths of the legs are 18 and 24,
  • The hypotenuse is 30.

Therefore, the correct response is:

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