Question

find the value of x.

x = ?

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Explanation:

Step1: Apply the Angle - Bisector Theorem

The Angle - Bisector Theorem states that if a ray bisects an angle of a triangle, then it divides the opposite side into segments proportional to the other two sides. So, \(\frac{6x + 6}{9x-9}=\frac{32}{24}\).

Step2: Simplify the proportion

Simplify \(\frac{32}{24}=\frac{4}{3}\). The equation becomes \(\frac{6x + 6}{9x-9}=\frac{4}{3}\). Cross - multiply: \(3(6x + 6)=4(9x - 9)\).

Step3: Expand both sides

Expand \(3(6x + 6)=18x+18\) and \(4(9x - 9)=36x-36\). So, \(18x + 18=36x-36\).

Step4: Solve for \(x\)

Subtract \(18x\) from both sides: \(18=18x-36\). Add \(36\) to both sides: \(18x=54\). Divide both sides by \(18\): \(x = 3\).

Answer:

\(3\)