Final answer:
To find AC, use the power theorem. Plug in the given values and solve for AC by moving all the terms involving AC to one side and simplifying the expression. The final value of AC is -2 + √3.
Step-by-step explanation:
To find AC, we can use the power theorem. The power theorem states that if A and B are points on a circle and AB is a chord of the circle, then the product of the lengths of the segments of the chord is equal. In this case, AB = 4√3 and AH = 4. So, using the power theorem, we have AB · AC = AH · AD, where AD is the length of the other segment of the chord. Plugging in the given values, we get 4√3 · AC = 4 · AD. Since AH = 4 and AD + AC = AB, we can substitute AD = AB - AC into the equation to get 4√3 · AC = 4 · (AB - AC). Solving for AC, we have 4√3 · AC = 4 · (4√3 - AC). Expanding and simplifying, we get 4√3 · AC = 16√3 - 4AC. Moving all the terms involving AC to one side, we get 4AC + 4√3 · AC = 16√3. Combining like terms, we have (4 + 4√3)AC = 16√3. Finally, dividing both sides by (4 + 4√3), we get AC = 16√3 / (4 + 4√3). Rationalizing the denominator, we get AC = 16√3 / (4 + 4√3) × (4 - 4√3) / (4 - 4√3), which simplifies to AC = (64 - 48√3) / (16 - 48) = (64 - 48√3) / (-32) = -2 + √3.