Question

1. \\( \triangle a b c \\) is an isosceles triangle with \\( \overline{a b} \cong \overline{b c} \\). if \\( a b=3 x + 41, b c = 8 x - 54 \\), and \\( a c = 12 x - 103 \\), find \\( x \\) and the measure of each side.

Explanation:

Step1: Set up the equation for equal sides

Since \(AB = BC\), we have \(3x + 41=8x - 54\).

Step2: Solve for \(x\)

Subtract \(3x\) from both sides: \(41 = 5x-54\).
Add \(54\) to both sides: \(41 + 54=5x\), so \(95 = 5x\).
Divide both sides by \(5\): \(x=\frac{95}{5}=19\).

Step3: Find the length of \(AB\)

Substitute \(x = 19\) into \(AB=3x + 41\): \(AB=3\times19 + 41=57 + 41 = 98\).

Step4: Find the length of \(BC\)

Substitute \(x = 19\) into \(BC = 8x-54\): \(BC=8\times19-54=152 - 54=98\).

Step5: Find the length of \(AC\)

Substitute \(x = 19\) into \(AC=12x - 103\): \(AC=12\times19-103=228-103 = 125\).

Answer:

\(x = 19\), \(AB = 98\), \(BC = 98\), \(AC=125\)