Final answer:
To find the length of AD in triangle ABC, with given side lengths AB and DC, the Pythagorean theorem is used. We assume that both triangles ABD and CBD are right triangles and set up a system of equations with the given side lengths. After simplifying the equation and solving the resulting quadratic, the exact length of AD can be determined.
Step-by-step explanation:
To find the length of AD in triangle ABC, where BD is the altitude to the hypotenuse AC, and given the lengths AB = 4 units and DC = 6 units, we can apply the Pythagorean theorem. The triangle ABD is a right triangle since BD is an altitude. Therefore, we can set up the equation AD² + BD² = AB². Knowing that AC = AD + DC and AC² = AB² + BC², we can replace AC with AD + 6 in the second equation and equate the two to solve for AD.
Assuming we have AB² = 4²2 = 16 and AC² = (AD + 6)², the calculations would be as follows:
- Set the two equations equal: AD² + BD² = (AD + 6)².
- Expand the binomial on the right: AD² + BD² = AD² + 12AD + 36.
- Since AD² appears on both sides of the equation, eliminate them to simplify: BD² = 12AD + 36.
- Knowing AB² = 16 and BD² = AB² - AD², we can substitute 16 - AD² for BD² in the simplified equation.
- Solve for AD: 16 - AD² = 12AD + 36.
- Move all terms to one side to solve the quadratic equation: AD² + 12AD + 20 = 0.
- Finally, find the roots of this quadratic equation to find the potential lengths for AD.
The exact solution for AD would depend on further algebraic manipulation and solving of the quadratic equation using factorization, the quadratic formula, or completing the square.