Calculating hypotenuse length using geometric mean relationships
In the given right triangle \( ABC \), with altitude \( BD \) drawn to the hypotenuse \( AC \), and \( BD = 6 \) and \( DC = 12 \), we can find the length of \( AC \) by using the geometric mean relationships that exist in a right triangle with an altitude drawn to its hypotenuse.
The length of \( AC \) is the sum of \( AD \) and \( DC \). We know \( DC = 12 \), but we need to find \( AD \) before we can find \( AC \).
The altitude \( BD \) is the geometric mean of the two segments of the hypotenuse. So we have \( BD^2 = AD \cdot DC \).
Given that \( BD = 6 \) and \( DC = 12 \), we plug these values into the equation:
\( 6^2 = AD \cdot 12 \)
\( 36 = AD \cdot 12 \)
Now, solve for \( AD \):
\( AD = \frac{36}{12} = 3 \)
Now that we know \( AD \) is 3 and \( DC \) is 12, we can find the length of \( AC \) by adding these two lengths:
\( AC = AD + DC = 3 + 12 = 15 \)
So, the length of the hypotenuse \( AC \) is 15 units.
