Question

solve for the variable x.
x = 2
x = 1
x = 10
x = 8.5

Explanation:

Step1: Apply the basic proportionality theorem (Thales' theorem)

If a line is drawn parallel to one side of a triangle intersecting the other two sides, then it divides those sides proportionally. So, \(\frac{12 - 3}{3}=\frac{2x - 3}{x - 5}\)

Step2: Simplify the left - hand side

\(\frac{9}{3}=\frac{2x - 3}{x - 5}\), which simplifies to \(3=\frac{2x - 3}{x - 5}\)

Step3: Cross - multiply

\(3(x - 5)=2x - 3\)

Step4: Expand the left - hand side

\(3x-15 = 2x - 3\)

Step5: Solve for \(x\)

Subtract \(2x\) from both sides: \(3x-2x-15=2x - 2x- 3\), so \(x-15=-3\). Then add \(15\) to both sides: \(x=-3 + 15\)

Answer:

\(x = 12\)

(It seems there might be an error in the provided options as the correct value of \(x\) based on the proportionality theorem is \(x = 12\). If we assume a different proportionality setup (maybe a mis - interpretation of the sides in the triangle - like \(\frac{12}{3}=\frac{2x - 3}{x - 5}\)):

Step1: Cross - multiply (for \(\frac{12}{3}=\frac{2x - 3}{x - 5}\))

\(12(x - 5)=3(2x - 3)\)

Step2: Expand

\(12x-60 = 6x - 9\)

Step3: Solve for \(x\)

Subtract \(6x\) from both sides: \(12x-6x-60=6x - 6x-9\), \(6x-60=-9\). Add \(60\) to both sides: \(6x=-9 + 60\), \(6x = 51\), \(x=\frac{51}{6}=8.5\))