In $\triangle ABC$, a line $DE$ is drawn parallel to the base $BC$, intersecting side $AB$ at point $D$ and side $AC$ at point $E$.If $AD = x$, $DB = x - 2$, $AE = x + 2$, and $EC = x - 1$, find the value of $x$ and the total length of side $AB$.

Step 1: Apply Basic Proportionality Theorem (BPT)

Since the line segment $DE$ is parallel to the base $BC$ in $\triangle ABC$, we can apply the Basic Proportionality Theorem (also known as Thales Theorem). This theorem states that if a line parallel to one side of a triangle intersects the other two sides, then it divides the two sides proportionally.

Step 2: Substitute the given values

Now, we substitute the given expressions for the lengths of the segments into the proportionality equation. We have $AD = x$, $DB = x - 2$, $AE = x + 2$, and $EC = x - 1$.

Step 3: Solve the equation for x

To solve for $x$, we cross-multiply the terms in the equation. This leads to a quadratic equation, which simplifies to a linear equation after cancelling out the $x^2$ terms. We then isolate $x$ to find its value.

Step 4: Calculate the lengths of the segments

Now that we have found the value of $x = 4$, we can substitute this value back into the expressions for the lengths of the segments to find their numerical values. This also helps verify that all lengths are positive, as required for geometric segments.

Step 5: Calculate the total length of side AB

The total length of side $AB$ is the sum of the lengths of its segments $AD$ and $DB$. We add the calculated numerical values to find the total length.

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