Final answer:
The length of AD in the given right triangle ABC with altitude BD is 12 units.
Step-by-step explanation:
Given right triangle ABC with altitude BD drawn to hypotenuse AC. If BD = 3 and DC = 9, we can find the length of AD using the Pythagorean theorem.
The Pythagorean theorem states that the square of the hypotenuse (AC) is equal to the sum of the squares of the other two sides (AB and BC). So, in this case, AC² = AB² + BC².
Since BD is the altitude, we can split triangle ABC into two smaller right triangles: ABD and BCD. So, AB is the height of triangle ABD and BC is the height of triangle BCD.
With the given information, we can write the equation: AC² = AB² + BC². Substituting the values, we have (AD + 3)² = (AB)² + (9)².
Expanding the equation: AD² + 6AD + 9 = AB² + 81.
Since AD and AB represent the same side length, we can simplify the equation to: AD² + 6AD + 9 = AD² + 81.
Now, subtracting AD² from both sides of the equation gives us 6AD + 9 = 81.
Next, subtracting 9 from both sides gives us 6AD = 72.
Finally, dividing both sides by 6 gives us AD = 12.
Therefore, the length of AD is 12 units.