To solve this problem, we must determine the length of the third side of a triangle where two sides are 24 and 22, and the angles are in Arithmetic Progression (AP).
Firstly, understand that if \(A\), \(B\), and \(C\) are the angles of a triangle, and they are in AP, then there exists a common difference \(d\) such that:
Since the sum of angles \(A + B + C = 180^\circ\), we have:
\(A + (A + d) + (A + 2d) = 180^\circ\)
Simplifying, we get:
\(3A + 3d = 180^\circ\)
Divide through by 3:
\(A + d = 60^\circ\)
This implies \(B = 60^\circ\).
Now, we can use the law of cosines to find the length of the third side.
The law of cosines states that for a triangle with sides \(a\), \(b\), and \(c\), and the angle \(C\) opposite side \(c\),
\(c^2 = a^2 + b^2 - 2ab \cos(C)\)
Assuming the sides opposite angles \(A\), \(B\), and \(C\) are 24, 22, and the third side \(c\) respectively, and since \(B = 60^\circ\), calculate:
\(c^2 = 24^2 + 22^2 - 2 \times 24 \times 22 \times \cos(60^\circ)\)
Given \(\cos(60^\circ) = \frac{1}{2}\), the equation becomes:
\(c^2 = 576 + 484 - 2 \times 24 \times 22 \times \frac{1}{2}\)
Simplifying further:
\(c^2 = 576 + 484 - 528\) \(c^2 = 532\)
Thus, solve for \(c\):
\(c = \sqrt{532}\)
Breaking it down:
\(532 = 4 \times 133 = 4 \times (144 - 11) = 4 \times (12^2 - 3^2)\)
Therefore,
\(c = \sqrt{4 \times (12^2 - 3^2)} = 2 \sqrt{12^2 - 3^2} = 2 \sqrt{144 - 9} = 2 \sqrt{135}\)
The simplified form is:
\(c = 12 + 2\sqrt{3}\)
Hence, the third side of the triangle is \(12 + 2\sqrt{3}\).
The correct answer is:
\(12 + 2\sqrt{3}\)