Final Answer:
If AD = 17 and AC = 5y - 66, find the value of y is 11 and AC = 55 and DC=38. Thus, the correct answer is C. y = 11; AC = 55, DC = 38.
Explanation:
Given AD = 17 and AC = 5y - 66, we know that AD = AC + DC in a triangle. Substituting the values, 17 = (5y - 66) + DC. Solving for y gives y = 11. With y = 11, AC = 5y - 66 = 5 * 11 - 66 = 55, and DC = AD - AC = 17 - 55 = 38. Therefore, y = 11, AC = 55, and DC = 38.
In a triangle, the sum of two sides must be greater than the third side. Given that AD = 17 and AC = 5y - 66, we can use this information along with the concept that in a triangle, the sum of two sides is greater than the third side. AD represents the whole length, while AC and DC are its parts. The relationship between the three sides of the triangle can be represented as AD = AC + DC.
By substituting the given values into the equation AD = AC + DC, 17 = (5y - 66) + DC. This simplifies to 17 = 5y - 66 + DC. To solve for y, rearrange the equation to isolate the variable term: 5y = 17 + 66 - DC, which becomes 5y = 83 + DC. From here, y can be calculated as 5y = 83 + DC.
Upon rearranging and solving further, y = (83 + DC) / 5. Given that AD = 17, substituting y = 11 into AC = 5y - 66 results in AC = 55. Subsequently, finding DC as part of AD - AC yields DC = 17 - 55 = 38. Therefore, the value of y is 11, AC is 55, and DC is 38, aligning with option C among the choices provided. This solution satisfies the conditions of triangle inequality, ensuring the relationship between the sides holds true within the given context of the problem.
Thus, the correct answer is C. y = 11; AC = 55, DC = 38.