Let ABC be right-angled triangle with ∠B = 90° . Let the length of the altitude BD be equal to 12. What is the minimum possible length of AC, given that AC and the perimeter of triangle ABC are integers?
Hint 1: Start by analyzing the initial conditions and setting up the basic equations. Let + = I Iin integer.
Hint 2: Look for algebraic properties, symmetry, or geometric theorems to simplify. a2 + c2 = I2 – 24b = b2.
Hint 3: Proceed with the final algebraic steps to solve the system. I2 = Ix solve for the final valueb.
Step 1: Let + = I Iin integer
Step 2: a2 + c2 = I2 – 24b = b2
Step 3: I2 = Ix = 12b
Step 4: I 2 − 48b 0
Step 5: b 2 + 24 l 0 24
Step 6: Let = – 12 36
Step 7: I2 = k2 – 144
Step 8: 144 = (k – I) (k + I)
Step 9: For = 37, I = 35
Step 10: I = 25 is minimum value
Step 11: ❑ ❑ ❑
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