Final answer:
To determine the length of the hypotenuse AC in the right triangle ABC, we apply the Pythagorean theorem with the given side lengths AD = 12 and BD = 24, which yields AC = 6√20 as the final answer.
Step-by-step explanation:
To find the length of the hypotenuse AC in right triangle ABC with altitude BD, we can use the Pythagorean theorem which states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b): c² = a² + b².
Given that AD = 12 and BD = 24, we can assign these values to the sides a and b in the Pythagorean theorem:
AC² = AD² + BD²
Plugging in the values gives us:
AC² = 12² + 24²
AC² = 144 + 576
AC² = 720
Therefore, AC = √720, which simplifies to AC = 6√20. This is the length of the hypotenuse AC.